One of the most essential methodologies for solving optimization problems is dynamic programming (DP), where the so-called principle of optimality, as defined by Bellman [1] in 1957, is used to create its methods. The multi-objective dynamic programming (MODP) is a method for resolving problems with competing objective functions that follows the DP properties (Mine and Fukushima [2], 1979; Carraway et al. [3]; Abo- Sinna and Hussein [4]; Abo-Sinna and Hussein [5]).

Osman [6-7] introduces the ideas of solvability set, the stability sets of first-kind and second-kind, as well as the analysis of these terms for parametric convex non-linear programming problem. In order to a certain class of multi-objective convex programming problems, Osman and Dauer [8] dedicated themselves to the discovering of the first-kind stability set. In addition, they provided a technique to compute this set and the related pareto optimum solution.

First and foremost, Zadeh [9] presented the philosophy of fuzziness in literature, which can be applied to deal with the issues in real-life scenario where the information is in the form of ambiguousness and incompleteness. Bellman and Zadeh [10] created a method for solving decision-making problems involving fuzziness that improved and aided managerial decision-making. Linear programming along with fuzzy programming involving numerous objective functions were presented by Zimmermann [11] in 1978. Several people afterwards worked in the field of fuzziness. As a result of the convenience, the piecewise linear fuzzy numbers such as interval, triangular, trapezoidal, pentagonal, hexagonal fuzzy numbers, etc. have been applied in the literature [12-16]. Many authors have investigated the solution methodology as well as their applications involving fuzziness, fuzzy systems, and fuzzy mathematical programming problems [17-18].

In the literature, fuzzy dynamic programming models in particular have gotten a lot of attention (see, Bellman and Zadeh [10]; Zimmermann [19]; Esogbue [20]; Esogbue and Bellman [21]; Hussein and Abo- Sinna [22]). Tanaka and Asai [23] introduced fuzzy parameters to multi-objective linear programming (MOLP) problems. General fuzzy multi-objective non-linear programming (MONLP) models were formulated by Orlovski [24] in 1984. Sakawa and Yano [25-26] developed the idea of pareto optimum optimality and proposed a new interactive fuzzy approach for MOLP and MONLP issues with fuzzy parameters. For fuzzy MONLP situations, Osman and El-Banna [27], in 1993, proposed a qualitative analysis and stability. There are enormous researches, who developed the MODP (for instance, Moghaddam and Ghoseiri [28]; Muruganantham et al. [29]; Li et al. [30]; Deng et al. [31]; Besheli et al. [32]; Peraza et al. [33]; Azevedo et al. [34]; Ni et al. [35]; Wu et al. [36]; Liu et al. [37]; Zou et al. [38]; and Zhang et al. [39]).

Based on aforesaid literature survey, in this article, the parameters of the model are re-defined and further studied for MODP problems involving the fuzzy parameters in objective functions, as a result of the above literature.

The main contribution of this article is as follows:

The remainder of this article is organized as in Figure 1 below:

In this section, we recall some basic concepts.

A minimization type problem involving the fuzzy parameters within the objective functions is formulated as follows

(PQF-VMP) $\min {\mathrm{G}}_{\mathrm{l}}\left({\mathrm{g}}_{\mathrm{l}1}\left({\mathrm{x}}_1,{\tilde{\mathrm{a}}}_1^{\mathrm{P}\mathrm{Q}}\right),{\mathrm{g}}_{\mathrm{l}2}\left({\mathrm{x}}_2,{\tilde{\mathrm{a}}}_2^{\mathrm{P}\mathrm{Q}}\right),\dots ,{\mathrm{g}}_{\mathrm{l}\mathrm{N}}\left({\mathrm{x}}_{\mathrm{N}},{\tilde{\mathrm{a}}}_{\mathrm{N}}^{\mathrm{P}\mathrm{Q}}\right)\right),\mathrm{l}=\overline{1,\mathrm{L}},\mathrm{L}\ge 2$s. t.

${\mathrm{x}}_{\mathrm{n}}\in {\mathrm{X}}_{\mathrm{n}},\mathrm{n}=\overline{1,\mathrm{N}}$.

Here, ${\mathrm{X}}_{\mathrm{n}}\subset {\mathfrak{R}}^{{\mathrm{r}}_{\mathrm{n}}},\mathrm{n}=\overline{1,\mathrm{N}};{\mathrm{x}}_{\mathrm{n}}$is a ${\mathrm{r}}_{\mathrm{n}}$ vector, ${\mathrm{G}}_{\mathrm{l}},\mathrm{l}=\overline{1,\mathrm{L}}$ and ${\mathrm{H}}_{\mathrm{q}},\mathrm{q}=\overline{1,\mathrm{Q}}$ are convex real function of class ${\complement }^{\left(1\right)}$ on ${\mathfrak{R}}^{\mathrm{N}}$ and ${\mathrm{g}}_{\mathrm{l}\mathrm{n}},{\mathrm{h}}_{\mathrm{q}\mathrm{n}}.\mathrm{l}=\overline{1,\mathrm{L}};\mathrm{q}=\overline{1,\mathrm{Q}};\mathrm{n}=\overline{1,\mathrm{N}}$ are real-valued functions on ${\mathrm{X}}_{\mathrm{n}}$, and ${\tilde{\mathrm{a}}}^{\mathrm{P}\mathrm{Q}}=\left({\tilde{\mathrm{a}}}_{11}^{\mathrm{P}\mathrm{Q}},{\tilde{\mathrm{a}}}_{22}^{\mathrm{P}\mathrm{Q}},\dots ,{\tilde{\mathrm{a}}}_{\mathrm{l}\mathrm{n}}^{\mathrm{P}\mathrm{Q}}\right),\mathrm{l}=\overline{1,\mathrm{L}};\mathrm{n}=\overline{1,\mathrm{N}}$ represent the fuzzy parameters in vector form, in ${\mathrm{f}}_{\mathrm{l}\mathrm{n}}\left({\mathrm{x}}_{\mathrm{n}},{\tilde{\mathrm{a}}}_{\mathrm{l}\mathrm{n}}^{\mathrm{P}\mathrm{Q}}\right).$ It is assumed that the aforesaid fuzzy parameters are designated as per the reference Jain [41], as well as the PQF-VMP is stable (Rockafellar [44]).

For a certain value of $\alpha ,$ the aforesaid PQF-VMP problem is converted into the following problem (Sakawa and Yano [25])

($\alpha$-VMP) $\min {\mathrm{G}}_{\mathrm{l}}\left({\mathrm{g}}_{\mathrm{l}1}\left({\mathrm{x}}_1,a_1\right),{\mathrm{g}}_{\mathrm{l}2}\left({\mathrm{x}}_2,{\mathrm{a}}_2\right),\dots ,{\mathrm{g}}_{\mathrm{l}\mathrm{N}}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{a}}_N\right)\right),\mathrm{l}=\overline{1,\mathrm{L}},\mathrm{L}\ge 2$

s. t.

Since PQF-VMP problem becomes stable, therefore the $\alpha$-VMP would also be stable.

Similarly, it can be illustrated that

In a situation when all the objectives as well as the constraints become separable, we claim that the $\alpha$-VMP problem would also be separable. In addition, the functions, designated by ${\mathsf{\varphi }}_{\mathrm{l}}^{\mathrm{n}}$ and ${\mathsf{\chi }}_{\mathrm{q}}^{\mathrm{n}}$, are referred as the separating functions for the set $\mathrm{G}$as well as for the set $\mathrm{H}$.

Consequently, the separation of the $\alpha$-VMP is termed as monotone provided that all ${\mathsf{\varphi }}_{\mathrm{l}}^{\mathrm{n}}$ and ${\mathsf{\chi }}_{\mathrm{q}}^{\mathrm{n}}$ are strictly increasing functions relative to the first argument for each constant second argument for each $\mathrm{y}\in \mathfrak{R}$,

Assumption 1: The $\alpha$-VMP problem possesses the reparability property. In addition, the separation property refers to monotonicity.

Assumption 2. For each $n,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}{\mathrm{X}}_{\mathrm{n}}\cap {\mathrm{L}}_{\mathsf{\alpha }}\left(\tilde{\mathrm{a}}\right)$ is assumed to be compact and ${\mathrm{G}}_{\mathrm{l}}^{\mathrm{n}}\left({\mathrm{g}}_{\mathrm{l}1}\left({\mathrm{x}}_1,{\mathrm{a}}_1\right),{\mathrm{g}}_{\mathrm{l}2}\left({\mathrm{x}}_2,{\mathrm{a}}_2\right),\dots ,{\mathrm{g}}_{\mathrm{l}\mathrm{n}}\left({\mathrm{x}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}}\right)\right),l=\overline{1,L}$. Also, we assume that ${\mathrm{H}}_{\mathrm{q}}^{\mathrm{n}}\left({\mathrm{h}}_{\mathrm{q}1}\left({\mathrm{x}}_1\right),{\mathrm{h}}_{\mathrm{q}2}\left({\mathrm{x}}_2\right),\dots ,{\mathrm{h}}_{\mathrm{Q}\mathrm{N}}\left({\mathrm{x}}_{\mathrm{N}}\right)\right),q=\overline{1,Q}$ are continuous functions of $\left(x_1,x_2,\dots ,x_n\right)$and $\left(a_1,a_2,\dots ,a_n\right)$.

Based on the weighting method (Chankong and Haimes [40]), $\alpha$-VMP problem can be treated as presented below: s. t.

Consider that each $G_l$ follows the addition rule, i. e., for $\mathrm{l}=\overline{1,\mathrm{L}}$, we have

${\mathrm{G}}_{\mathrm{l}}\left({\mathrm{g}}_{\mathrm{l}1}\left({\mathrm{x}}_1,{\mathrm{a}}_1\right),{\mathrm{g}}_{\mathrm{l}2}\left({\mathrm{x}}_2,{\mathrm{a}}_2\right),\dots ,{\mathrm{g}}_{\mathrm{l}\mathrm{N}}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{a}}_{\mathrm{n}\mathrm{N}}\right)\right)={\overline{\mathrm{g}}}_{\mathrm{l}1}\left({\mathrm{x}}_1,{\mathrm{a}}_1\right),{\overline{\mathrm{g}}}_{\mathrm{l}2}\left({\mathrm{x}}_2,{\mathrm{a}}_2\right),\dots ,{\overline{\mathrm{g}}}_{\mathrm{L}\mathrm{N}}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{a}}_{\mathrm{n}\mathrm{N}}\right)$.

So, the objective function in $\alpha$-VMP_w attaints the following form:

Now, let us define:

The recursive relation, for $\mathrm{n}=\overline{1,\mathrm{N}}$ is presented as follows: where, ${\mathrm{u}}^{n-1}\left(x_n,u\right)=\left({\mathrm{u}}_1^{n-1}\left(x_n,u\right),\dots ,{\mathrm{u}}_Q^{n-1}\left(x_n,u\right)\right).$Assuming the monotonicity of ${\mathrm{G}}_{\mathrm{l}}^{\mathrm{l}}$, let ${\mathrm{u}}_Q^{n-1}$ be defined as

Theorem 1. Let the assumptions 1 and 2 be satisfied. Also, suppose $\left(x_1^{*},x_2^{*},\dots ,x_n^{*}\right),$$\left({\mathrm{a}}_1^{\mathrm{*}},{\mathrm{a}}_2^{\mathrm{*}},\dots ,{\mathrm{a}}_{\mathrm{n}}^{\mathrm{*}}\right){\in \mathrm{L}}_{\mathsf{\alpha }}\left(\tilde{\mathrm{a}}\right)$ is an $\alpha -$pareto optimal solution of ${\mathrm{A}}_{\mathrm{n}}\left({\mathrm{w}}^{*},\mathrm{u}\right)$ for some ${\mathrm{w}}^{\mathrm{*}}\in \mathrm{W}$. Then $\left({\mathrm{x}}_1^{\mathrm{*}},{\mathrm{x}}_2^{\mathrm{*}},\dots ,{\mathrm{x}}_{\mathrm{n}-1}^{\mathrm{*}}\right),\left({\mathrm{a}}_1^{\mathrm{*}},{\mathrm{a}}_2^{\mathrm{*}},\dots ,{\mathrm{a}}_{\mathrm{n}-1}^{\mathrm{*}}\right){\in \mathrm{L}}_{\mathsf{\alpha }}\left(\tilde{\mathrm{a}}\right)$ would be an $\mathsf{\alpha }-$pareto optimal solution for${\mathrm{A}}_{\mathrm{n}-1}\left({\mathrm{w}}^{\mathrm{*}},{\mathrm{u}}^{\mathrm{n}-1}({\mathrm{x}}_{\mathrm{n}},\mathrm{u})\right).$

Proof (see, Mine and Fukushima [2]).

Definition 9. Given a particular ${\mathrm{w}}^{\mathrm{*}}$ containing the corresponding $\alpha -$pareto optimal solution$\left({\mathrm{x}}^{\mathrm{*}},{\mathrm{a}}^{\mathrm{*}}\right)$. The stability set of first kind of $\left(\alpha -\mathrm{V}\mathrm{M}\mathrm{P}\right)$ relative to $\left({\mathrm{x}}^{\mathrm{*}},{\mathrm{a}}^{\mathrm{*}}\right)$ is designated as follows:

In this section, an algorithm for determining the $\mathrm{S}\left({\mathrm{x}}^{\mathrm{*}},{\mathrm{a}}^{\mathrm{*}}\right)$ is presented in the below steps:

Step 1: Start at $\alpha =0.$

Step 2: Define the membership grades of the fuzzy number $\tilde{a}$ as per definition 2.

Step 3: Formulate the piecewise quadratic fuzzy dynamic multi-objective problem,

i. e., (PQF-VMP)

Step 4: Choose$\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}{\mathrm{w}}^{*}\in W$, that is by using the relation (2) to achieve the $\alpha -$pareto optimal solution $\left({\mathrm{x}}^{\mathrm{*}},{\mathrm{a}}^{\mathrm{*}}\right)$ of ($\alpha$-VMP_w).

Step 5: Putting the value of the $\left({\mathrm{x}}^{\mathrm{*}},{\mathrm{a}}^{\mathrm{*}}\right)$ in the Kuhn- Tucker conditions, we have systems (11) as well as (15). In addition, we can use the Gauss Elimination method for solving system (12).

Step 6: Based on the Lagrange multipliers values, we obtain

Step 7: Set $\alpha =\left(\alpha +\epsilon \right)\in \left[0,1\right]$. Then, move to step 1.

Step 8: Repeat the interval at the steps of the proposed algorithm until the $\left[0,1\right]$ is completely nullified.

Take into account the following (PQF-VMP) s. t.

Here, ${\tilde{\mathrm{a}}}_{11}^{\mathrm{p}\mathrm{q}}=\left(0,0.2,3,\mathrm{4,6},6\right),{\tilde{\mathrm{a}}}_{22}^{\mathrm{P}\mathrm{Q}}=\left(0.2,0.4,4,5.6,7\right),{\tilde{\mathrm{a}}}_{33}^{\mathrm{P}\mathrm{Q}}=(2,3.4,6,9.8,11)$The close intervals approximation for ${\tilde{\mathrm{a}}}_{11}^{\mathrm{p}\mathrm{q}},{\tilde{\mathrm{a}}}_{22}^{\mathrm{P}\mathrm{Q}}$, and ${\tilde{\mathrm{a}}}_{33}^{\mathrm{P}\mathrm{Q}}$ are as follows:

The $(0.5$-VMP) can be written as s. t.