New Approach for Investigating a Class of Multi-Cost Interval-Valued Extremization Problems

Abstract

This study concentrates on a new approach for solving a class of multi-cost convex interval-valued extremization problems. Namely, we apply the weighting technique to find efficient solutions to these problems, defined in terms of $LU$-efficiency and weak $LU$-efficiency. Thus, an auxiliary weighting extremization problem related to the considered multi-cost interval-valued extremization problem is introduced. Under appropriate convexity hypotheses, an equivalence is established between the (weakly) $LU$-efficient solution of the multi-cost interval-valued extremization problem and the optimal solution of the auxiliary weighting extremization problem. Also, a numerical example is formulated to support the theoretical developments derived in the paper.

1. Introduction

In operations research, extremum problems are typically addressed under the assumption that all components are deterministic real numbers. However, real-world applications across various fields such as industry, engineering, physics, machine learning, data analytics, finance and risk management, medicine, and control theory often involve data that are imprecise and ambiguous. This necessitates optimization problems where the parameters are not precisely known, introducing inherent uncertainty in the modelling process. Several methodologies exist to investigate and solve uncertain optimization problems, interval optimization being one of them. Instead of real numbers, intervals are used to represent uncertainty. Interval-valued optimization problems specifically address situations where uncertain parameters are expressed as closed intervals in both the objective functions and constraints. Significant advancements in this area were made by Moore [1,2,3] by introducing foundational concepts and methods in interval analysis. Alfred and Herzberger [4] focused on the theory and practice of interval programming, which is used to ensure that uncertainty and error margins are permanent in extended programming, along with the basic principle and summary of interval programming. Charnes et al. [5] explored linear programming problems with constraints represented by closed intervals, while Ishibuchi and Tanaka [6] dealt with interval-valued objective functions without interval uncertainty in the constraints. Due to the development of interval analysis, many researchers (see, for example, [7,8,9,10,11,12,13]) have proposed various approaches to solve interval-valued optimization problems. Pereira [14] explained how the control design system for autonomous vehicles can be developed using optimization methods.

The concept of invexity, especially interval-valued symmetric invexity, plays a vital role in solving interval optimization problems. Guo et al. [15] developed new methods for addressing non-smooth interval optimization problems based on this concept. Moreover, Saeed and Treanţă [16] contributed significantly by introducing new classes of interval-valued variational problems and inequalities. Recently, Antczak [9] applied the weighting method to nonlinear vector optimization problems with multiple interval-valued objectives, demonstrating that an optimal solution to the associated scalar weighting problem is equivalent to a weak Pareto solution for the original nonlinear interval-valued multi-objective programming problem under appropriate convexity assumptions. Other important studies in the literature have focused on interval-valued variational problems and their optimality conditions. Debnath and Pokharna [17] analyzed the optimality and duality in interval-valued variational problems with B-$(p,r)$-invexity. In this context, researchers like Giannessi [18] and Giorgi [19] have provided crucial insights into theorems of the alternative and optimality conditions. Treanţă [20,21] defined KT-pseudoinvex variational control problems with interval values and investigated the connections between $LU$-optimal solutions and the saddle points associated with interval-valued Lagrange functionals. Treanţă and Ciontescu [22] made a notable contribution by studying optimal control problems with generalized invariant convex interval-valued functionals. Additionally, they explored the connections between such problems and related inequalities in variational control problems (see [23]). Other connected ideas on this topic can be read in Kumar and Yao [24] and Tang and Hua [25].

Motivated by all these works, in this paper, we employ the weighting technique for a new family of multiple cost interval-valued control problems. Namely, by constructing an associated scalar weighting extremization problem and applying appropriate convexity hypotheses, we establish the equivalence between weakly $LU$-efficient solutions and $LU$-efficient solutions in the original problem and the associated scalar weighting extremization problem. Moreover, these results are illustrated with a suitable example of a convex two-cost interval-valued extremization problem. Based on this illustrative example, performing a comparative analysis of the results obtained using the weighting technique versus traditional optimization methods, we can easily conclude the effectiveness of the weighting technique in solving multiple cost optimization problems, that is, the conversion of a vector problem to a scalar one.

2. Preliminary Results

This section focuses on basic definitions, notations, and basic calculus in interval analyses, which are used in the following sections. In this regard, let ${\mathbb{R}}^r$ be the classical Euclidean space of dimension r, together with its non-negative orthant, denoted by ${\mathbb{R}}_{+}^r$. For any vectors $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_r\right)}^T$, and $\rho ={\left({\rho }_1,{\rho }_2,\dots ,{\rho }_r\right)}^T$ in ${\mathbb{R}}^r$ we define:

(i)

$\varrho =\rho ⟺{\varrho }_l={\rho }_l,\forall l\in {\Gamma }_r$;

(ii)

$\varrho <\rho ⟺{\varrho }_l<{\rho }_l,\forall l\in {\Gamma }_r$;

(iii)

$\varrho \leqq \rho ⟺{\varrho }_l\le {\rho }_l,\forall l\in {\Gamma }_r$;

(iv)

$\varrho \le \rho ⟺\varrho \leqq \rho$ and $\varrho \ne \rho$.

where ${\Gamma }_r=\{1,2,\dots ,r\}$ is an index set. We consider $\mathcal{J}$ as a compact real interval, K as a family of piecewise smooth functions $z:\mathcal{J}\to {\mathbb{R}}^n$ (state variable), and Y as the family of all piecewise continuous functions $y:\mathcal{J}\to {\mathbb{R}}^s$ (control variables). Furthermore, we consider $\phi :\mathcal{J}\times K\times Y\to {\mathbb{R}}^r,g:\mathcal{J}\times K\times Y\to {\mathbb{R}}^m$ and $h:\mathcal{J}\times K\times Y\to {\mathbb{R}}^n$ as vector-valued functionals that possess continuous differentiability concerning each of their inputs. The functional $\phi =\phi (ϵ,z(ϵ),y(ϵ\left)\right)$ is defined for an independent variable $ϵ\in \mathcal{J}$, with $z:\mathcal{J}\to {\mathbb{R}}^n$ as a n-dimensional piecewise smooth function of $ϵ$ ($\dot{z}\left(ϵ\right)$, which denotes the derivative of $z\left(ϵ\right)$ with respect to $ϵ$), and $y:\mathcal{J}\to {\mathbb{R}}^s$ as a s-dimensional piecewise continuous function of $ϵ$. To make the notation less complex, we will denote $z\left(ϵ\right),y\left(ϵ\right)$, and $\dot{z}\left(ϵ\right)$ as $z,y$, and $\dot{z}$, respectively. If ${\phi }^l,l=1,\dots ,r$, are the components of the above-mentioned vector-valued function $\phi$, the partial derivatives of ${\phi }^l$ with respect to $ϵ,z$, and y are denoted as ${\phi }_{ϵ}^l,{\phi }_z^l$, and ${\phi }_y^l$, respectively. More precisely, ${\phi }_z^l$ and ${\phi }_y^l$ are defined as the vectors ${\left({\displaystyle \frac{\mathsf{\partial }{\phi }^l}{\mathsf{\partial }{z}_1}},\dots ,{\displaystyle \frac{\mathsf{\partial }{\phi }^l}{\mathsf{\partial }{z}_n}}\right)}^T$ and ${\left({\displaystyle \frac{\mathsf{\partial }{\phi }^l}{\mathsf{\partial }{y}_1}},\dots ,{\displaystyle \frac{\mathsf{\partial }{\phi }^l}{\mathsf{\partial }{y}_s}}\right)}^T$, respectively. Similarly, the first-order partial derivatives $g_z,h_z$, and $g_y,h_y$ of the vector-valued functionals g and h, respectively, can be expressed using matrices with m/n rows instead of a single row.

Consider the family $\mathcal{I}\left(\mathbb{R}\right)$ of all compact (bounded and closed) real intervals. Further, when we refer to a compact real interval, we represent it as $\mathcal{R}=\left[{\underline{r}}^L,{\overline{r}}^U\right]$, where ${\underline{r}}^L$ and ${\overline{r}}^U$ represent the lower and upper bounds of $\mathcal{R}$, respectively. To clarify, if $\mathcal{R}=\left[{\underline{r}}^L,{\overline{r}}^U\right]\in \mathcal{I}\left(\mathbb{R}\right)$, then $\mathcal{R}=\left[{\underline{r}}^L,{\overline{r}}^U\right]=\left\{ϵ\in \mathbb{R}:{\underline{r}}^L\le ϵ\le {\overline{r}}^U\right\}$. If ${\underline{r}}^L={\overline{r}}^U=r$, then $\mathcal{R}=[r,r]=r$ is a real number. Let $\mathcal{R}=\left[{\underline{r}}^L,{\overline{r}}^U\right]$, and $\mathcal{P}=\left[{\underline{p}}^L,{\overline{p}}^U\right]$ in $\mathcal{I}\left(\mathbb{R}\right)$. Then, by definition, we have:

(a)

$\mathcal{R}+\mathcal{P}=\{r+p:r\in \mathcal{R}$ and $p\in \mathcal{P}\}=\left[{\underline{r}}^L+{\underline{p}}^L,{\overline{r}}^U+{\overline{p}}^U\right]$,

(b)

$-\mathcal{R}=\{-r:r\in \mathcal{R}\}=\left[-{\overline{r}}^U,-{\underline{r}}^L\right]$,

(c)

$\mathcal{R}-\mathcal{P}=\mathcal{R}+(-\mathcal{P})=\{r-p:r\in \mathcal{R}$ and $p\in \mathcal{P}\}=\left[{\underline{r}}^L-{\overline{p}}^U,{\overline{r}}^U-{\underline{p}}^L\right]$,

(d)

$\zeta +\mathcal{R}=\{\zeta +r:r\in \mathcal{R}\}=\left[\zeta +{\underline{r}}^L,\zeta +{\overline{r}}^U\right]$, where $\zeta$ is a real number,

(e)

$\zeta \mathcal{R}=\left\{\begin{array}{c}\left[\zeta {\underline{r}}^L,\zeta {\overline{r}}^U\right]\mathrm{if}\zeta >0,\hfill \\ \\ \left[\zeta {\overline{r}}^U,\zeta {\underline{r}}^L\right]\mathrm{if}\zeta \leqq 0,\hfill \end{array}\right.$ where $\zeta$ is a real number.

Interval analysis (see, for example, Moore [1,2], Moore et al. [3], and Alefeld and Herzberger [4]) commonly uses an order relation to establish a ranking among real intervals. This fact means that one interval is superior to another, but it does not imply that one is larger than the other. Thus, for $\mathcal{R}=\left[{\underline{r}}^L,{\overline{r}}^U\right]$, and $\mathcal{P}=\left[{\underline{p}}^L,{\overline{p}}^U\right]$ in $\mathcal{I}\left(\mathbb{R}\right)$, we write

$$\mathcal{R}{\le }_{LU}\mathcal{P}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\left\{\begin{array}{c}{\underline{r}}^L\le {\underline{p}}^L\hfill \\ \\ {\overline{r}}^U\le {\overline{p}}^U.\hfill \end{array}\right.$$

The fact that ${\le }_{LU}$ is a partial ordering on $\mathcal{I}\left(\mathbb{R}\right)$ is readily apparent. This implies that $\mathcal{R}$ is inferior to $\mathcal{P}$, or $\mathcal{P}$ is superior to $\mathcal{R}$. Moreover, $\mathcal{R}{<}_{LU}\mathcal{P}$ can be expressed if and only if $\mathcal{R}{\le }_{LU}\mathcal{P}$ and $\mathcal{R}\ne \mathcal{P}$, or, equivalently

$$\mathcal{R}{<}_{LU}\mathcal{P}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\left\{\begin{array}{c}{\underline{r}}^L<{\underline{p}}^L\hfill \\ \\ {\overline{r}}^U\le {\overline{p}}^U,\hfill \end{array}\mathrm{or}\left\{\begin{array}{c}{\underline{r}}^L\le {\underline{p}}^L\hfill \\ \\ {\overline{r}}^U<{\overline{p}}^U,\hfill \end{array}\mathrm{or}\left\{\begin{array}{c}{\underline{r}}^L<{\underline{p}}^L\hfill \\ \\ {\overline{r}}^U<{\overline{p}}^U.\hfill \end{array}\right.\right.\right.$$

A vector of compact real intervals, denoted as $\mathcal{R}=\left({\mathcal{R}}_1,\dots ,{\mathcal{R}}_p\right)$, is defined as a vector where each component ${\mathcal{R}}_l$ is a compact real interval $\left[{\underline{r}}_l^L,{\overline{r}}_l^U\right]$. Denote using $\mathcal{I}\left({\mathbb{R}}^p\right)$ the family of all vectors of compact real intervals. For two vectors of the compact real intervals, $\mathcal{R}=\left({\mathcal{R}}_1,\dots ,{\mathcal{R}}_p\right)$, and $\mathcal{P}=\left({\mathcal{P}}_1,\dots ,{\mathcal{P}}_p\right)$, we will use the notation $\mathcal{R}{\le }_{LU}\mathcal{P}$ to indicate that ${\mathcal{R}}_l{\le }_{LU}{\mathcal{P}}_l$, for all $l\in {\Gamma }_p$. Similarly, we will use $\mathcal{R}{<}_{LU}\mathcal{P}$ to indicate that ${\mathcal{R}}_l{\le }_{LU}{\mathcal{P}}_l$, for all $l\in {\Gamma }_p$ and ${\mathcal{R}}_{l^{*}}{<}_{LU}{\mathcal{P}}_{l^{*}}$ for at least one $l^{*}\in {\Gamma }_p$.

Now, we will formulate the convexity concept associated with an interval-valued controlled functional. In particular, we use the very straightforward concept of convexity introduced by Wu [26].

Definition 1.

Let $\Phi :K\times Y\to \mathcal{I}\left(\mathbb{R}\right)$ be defined by

$${\displaystyle \Phi (z,y)={\int }_a^b\phi (ϵ,z,\dot{z},y)dϵ=\left[{\int }_a^b{\phi }^L(ϵ,z,\dot{z},y)dϵ,{\int }_a^b{\phi }^U(ϵ,z,\dot{z},y)dϵ\right],}$$

with $\phi :\mathcal{J}\times K\times K\times Y\to \mathcal{I}\left(\mathbb{R}\right)$ and ${\phi }^L,{\phi }^U:\mathcal{J}\times K\times K\times Y\to \mathbb{R}$ continuously differentiable functionals. Then, ${\displaystyle \Phi (z,y)={\int }_a^b\phi (ϵ,z,\dot{z},y)dϵ}$ is said to be a convex interval-valued controlled functional on $K\times Y$ if the inequalities

$$\begin{array}{c}{\int }_a^b{\phi }^L(ϵ,z,\dot{z},y)dϵ-{\int }_a^b{\phi }^L(ϵ,\overline{z},\dot{\overline{z}},\overline{y})dϵ\ge \\ {\int }_a^b\left\{{(z-\overline{z})}^T{\phi }_z^L(ϵ,\overline{z},\dot{\overline{z}},\overline{y})+{(\dot{z}-\dot{\overline{z}})}^T{\phi }_{\dot{z}}^L(ϵ,\overline{z},\dot{\overline{z}},\overline{y})+{(y-\overline{y})}^T{\phi }_y^L(ϵ,\overline{z},\dot{\overline{z}},\overline{y})\right\}dϵ\\ {\int }_a^b{\phi }^U(ϵ,z,\dot{z},y)dϵ-{\int }_a^b{\phi }^U(ϵ,\overline{z},\dot{\overline{z}},\overline{y})dϵ\ge \\ {\int }_a^b\left\{{(z-\overline{z})}^T{\phi }_z^U(ϵ,\overline{z},\dot{\overline{z}},\overline{y})+{(\dot{z}-\dot{\overline{z}})}^T{\phi }_{\dot{z}}^U(ϵ,\overline{z},\dot{\overline{z}},\overline{y})+{(y-\overline{y})}^T{\phi }_y^U(ϵ,\overline{z},\dot{\overline{z}},\overline{y})\right\}dϵ\end{array}$$

hold for all $z,\overline{z}\in K,y,\overline{y}\in Y$.

The alternative lemma presented below is a specific case of the more general results established in Giannessi [18] and Giorgi [19] for convex vector optimization problems.

Lemma 1.

Let $K\times Y$ be a convex set and $\varphi :\mathcal{J}\times K\times Y\to \mathcal{I}\left({\mathbb{R}}^p\right)$ and $\psi :\mathcal{J}\times K\times Y\to \mathcal{I}\left({\mathbb{R}}^q\right)$ be convex interval-valued controlled functionals. If the system

$$\begin{array}{cc}\hfill {\int }_a^b{\varphi }_l(ϵ,z,y)dϵ=& \left[{\int }_a^b{\varphi }_l^L(ϵ,z,y)dϵ,{\int }_a^b{\varphi }_l^U(ϵ,z,y)dϵ\right]{<}_{LU}[0,0]\forall l\in {\Gamma }_p\hfill \\ \hfill {\int }_a^b{\psi }_j(ϵ,z,y)dϵ=& \left[{\int }_a^b{\psi }_j^L(ϵ,z,y)dϵ,{\int }_a^b{\psi }_j^U(ϵ,z,y)dϵ\right]{\le }_{LU}[0,0],\forall j\in {\Gamma }_q\hfill \end{array}$$

has no solution, then there exist $\delta =\left({\delta }^L,{\delta }^U\right)\ge 0$, where ${\delta }^L,{\delta }^U\in {\mathbb{R}}^p$, and $\xi =\left({\xi }^L,{\xi }^U\right)\ge 0$, where ${\xi }^L,{\xi }^U\in {\mathbb{R}}^q$, such that

$$\begin{array}{cc}& {\int }_a^b\left(\sum _{l=1}^p\left({\delta }_l^L{\varphi }_l^L(ϵ,z,y)+{\delta }_l^U{\varphi }_l^U(ϵ,z,y)\right)\right)dϵ\hfill \\ & +{\int }_a^b\left(\sum _{j=1}^q\left({\xi }_j^L{\psi }_j^L(ϵ,z,y)+{\xi }_j^U{\psi }_j^U(ϵ,z,y)\right)\right)dϵ\ge 0.\hfill \end{array}$$

3. Main Results

In this section, we investigate the following multi-cost interval-valued extremization problem formulated as follows

$$\begin{array}{c}\left(\mathrm{P}\right)\underset{(z,y)}{min}{\int }_{\mathcal{J}}\phi (ϵ,z,y)dϵ=\underset{(z,y)}{min}\left({\int }_{\mathcal{J}}{\phi }_1(ϵ,z,y)dϵ,\dots ,{\int }_{\mathcal{J}}{\phi }_r(ϵ,z,y)dϵ\right)\\ \mathrm{subject}\mathrm{to}g(ϵ,z,y)\leqq 0,\forall ϵ\in \mathcal{J}=[a,b],\\ h(ϵ,z,y)=\dot{z},\forall ϵ\in \mathcal{J},\\ z\left(a\right)=a_0,z\left(b\right)=b_0,\end{array}$$

where ${\displaystyle {\int }_{\mathcal{J}}{\phi }_l(ϵ,z,y)dϵ=\left[{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,z,y)dϵ,{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,z,y)dϵ\right]}$ for each $l\in {\Gamma }_r$, $a_0,b_0\in {\mathbb{R}}^n$ are given, and ${\phi }_l^L,{\phi }_l^U:I\times K\times Y\to \mathbb{R},l\in {\Gamma }_r,g:I\times K\times Y\to {\mathbb{R}}^m$ and $h:I\times K\times Y\to {\mathbb{R}}^n$ are $C^1$-class functionals. Let

$$\mathcal{A}=\{(z,y)\in K\times Y:g(ϵ,z,y)\leqq 0,h(ϵ,z,y)=\dot{z},\forall ϵ\in \mathcal{J},z\left(a\right)=a_0,z\left(b\right)=b_0\}$$

be the set of all feasible solutions in (P).

The optimal solutions for multiple objective interval-valued extremization problems are defined in terms of weakly $LU$-efficient and $LU$-efficient solutions, as below.

Definition 2.

A feasible solution $(\overline{z},\overline{y})\in \mathcal{A}$ is said to be a weakly $LU$-efficient solution of (P) if and only if there exists no other $(z,y)\in \mathcal{A}$ such that

$${\int }_{\mathcal{J}}{\phi }_l(ϵ,z,y)dϵ{<}_{LU}{\int }_{\mathcal{J}}{\phi }_l(ϵ,\overline{z},\overline{y})dϵ,\forall l\in {\Gamma }_r.$$

Definition 3.

A feasible solution $(\overline{z},\overline{y})\in \mathcal{A}$ is said to be a $LU$-efficient solution of (P) if and only if there exists no other $(z,y)\in \mathcal{A}$ such that

$$\begin{array}{c}{\displaystyle {\int }_{\mathcal{J}}}{\phi }_l(ϵ,z,y)dϵ{\le }_{LU}{\displaystyle {\int }_{\mathcal{J}}}{\phi }_l(ϵ,\overline{z},\overline{y})dϵ,\forall l\in {\Gamma }_r\\ {\int }_{\mathcal{J}}{\phi }_{i_0}(ϵ,z,y)dϵ{<}_{LU}{\int }_{\mathcal{J}}{\phi }_{i_0}(ϵ,\overline{z},\overline{y})dϵ,\mathit{for}\mathit{some}{i}_0\in {\Gamma }_r.\end{array}$$

In this section, to investigate a weakly $LU$-efficient solution and/or a $LU$-efficient solution of (P), we use the weighting approach (see Antczak [9]). Thus, for this purpose, an auxiliary weighting control problem is introduced for the considered multi-cost interval-valued extremization problem as follows:

$${(weight-P)}_v\underset{(z,y)}{min}\Pi (z,y)=\underset{(z,y)}{min}{\int }_{\mathcal{J}}\left\{\sum _{l=1}^r{v}_l^L{\phi }_l^L(ϵ,z,y)+\sum _{l=1}^r{v}_l^U{\phi }_l^U(ϵ,z,y)\right\}dϵ$$

$$\begin{array}{c}\mathrm{subject}\mathrm{to}g(ϵ,z,y)\leqq 0,\forall ϵ\in \mathcal{J},\\ h(ϵ,z,y)=\dot{z},\forall ϵ\in \mathcal{J},\\ z\left(a\right)=a_0,z\left(b\right)=b_0,\end{array}$$

where $v=(v^L,v^U)$, with $v^L=\left(v_1^L,\dots ,v_r^L\right)\ge 0$, $v^U=\left(v_1^U,\dots ,v_r^U\right)\ge 0$.

Definition 4.

A feasible solution $(\overline{z},\overline{y})\in \mathcal{A}$ is said to be an optimal solution of ${(weight-P)}_v$ if the inequality

$$\begin{array}{cc}& {\int }_{\mathcal{J}}\left\{\sum _{l=1}^r{v}_l^L{\phi }_l^L(ϵ,\overline{z},\overline{y})+\sum _{l=1}^r{\overline{v}}_l^U{\phi }_l^U(ϵ,\overline{z},\overline{y})\right\}dϵ\hfill \\ & \le {\int }_{\mathcal{J}}\left\{\sum _{l=1}^r{v}_l^L{\phi }_l^L(ϵ,z,y)+\sum _{l=1}^r{v}_l^U{\phi }_l^U(ϵ,z,y)\right\}dϵ\hfill \end{array}$$

holds for all $(z,y)\in \mathcal{A}$.

Theorem 1.

Let $(\overline{z},\overline{y})\in \mathcal{A}$ be an optimal solution of ${(weight-P)}_{\overline{v}}$. Further, assume that $\overline{v}=\left({\overline{v}}^L,{\overline{v}}^U\right)=\left(\left({\overline{v}}_1^L,{\overline{v}}_2^L,\dots ,{\overline{v}}_r^L\right),\left({\overline{v}}_1^U,{\overline{v}}_2^U,\dots ,{\overline{v}}_r^U\right)\right)\ge 0$ with $\left({\overline{v}}_{i_0}^L,{\overline{v}}_{i_0}^U\right)>0$ for some $i_0\in {\Gamma }_r$. Then, $(\overline{z},\overline{y})\in \mathcal{A}$ is a weakly $LU$-efficient solution of (P).

Proof. 

By assumption, $(\overline{z},\overline{y})\in \mathcal{A}$ is an optimal solution of ${(weight-P)}_{\overline{v}}$. We assume, on the contrary, that $(\overline{z},\overline{y})\in \mathcal{A}$ is not a weakly $LU$-efficient solution to (P). Therefore, according to Definition 2, it follows that there exists another $(\tilde{z},\tilde{y})\in \mathcal{A}$ such that

$${\int }_{\mathcal{J}}{\phi }_l(ϵ,\tilde{z},\tilde{y})dϵ{<}_{LU}{\int }_{\mathcal{J}}{\phi }_l(ϵ,\overline{z},\overline{y})dϵ,\forall l\in {\Gamma }_r.$$

According to the definition of the order relation ${<}_{LU}$, it follows that, for any $l\in {\Gamma }_r$, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ\le {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ,\hfill \end{array}\right.\\ \\ \hfill \mathrm{or}\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ\le {\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ,\hfill \end{array}\right.\\ \\ \hfill \mathrm{or}\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ.\hfill \end{array}\right.\end{array}$$

Since $\overline{v}=\left({\overline{v}}^L,{\overline{v}}^U\right)=\left(\left({\overline{v}}_1^L,\dots ,{\overline{v}}_r^L\right),\left({\overline{v}}_1^U,\dots ,{\overline{v}}_r^U\right)\right)\ge 0$ with $\left({\overline{v}}_{i_0}^L,{\overline{v}}_{i_0}^U\right)>0$ for some $i_0\in {\Gamma }_r$, the above system of inequalities yields that the inequality

$$\begin{array}{c}{\int }_{\mathcal{J}}\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{\phi }_l^L(ϵ,\tilde{z},\tilde{y})+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{\phi }_l^U(ϵ,\tilde{z},\tilde{y})\right\}dϵ<\\ {\int }_{\mathcal{J}}\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{\phi }_l^L(ϵ,\overline{z},\overline{y})+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{\phi }_l^U(ϵ,\overline{z},\overline{y})\right\}dϵ\end{array}$$

holds. This contradicts the assumption that $(\overline{z},\overline{y})\in \mathcal{A}$ is an optimal solution of ${(weight-P)}_{\overline{v}}$. Hence, $(\overline{z},\overline{y})\in \mathcal{A}$ is a weakly $LU$-efficient solution of the considered control problem (P), which completes the proof of the Theorem 1. □

Theorem 2.

Let $(\overline{z},\overline{y})\in \mathcal{A}$ be an optimal solution of ${(weight-P)}_{\overline{v}}$. Further, assume that $\overline{v}=\left({\overline{v}}^L,{\overline{v}}^U\right)=\left(\left({\overline{v}}_1^L,\dots ,{\overline{v}}_r^L\right),\left({\overline{v}}_1^U,\dots ,{\overline{v}}_r^U\right)\right)\ge 0$ with ${\overline{v}}^L\ge 0$ and ${\overline{v}}^U\ge 0$. Then, $(\overline{z},\overline{y})\in \mathcal{A}$ is an $LU$-efficient solution of (P).

Proof. 

By assumption, $(\overline{z},\overline{y})\in \mathcal{A}$ is an optimal solution of ${(weight-P)}_{\overline{v}}$. We assume, contrary to the result, that $(\overline{z},\overline{y})\in \mathcal{A}$ is not an $LU$-efficient solution to (P). Therefore, according to Definition 3, it follows that there exists another $(\tilde{z},\tilde{y})\in \mathcal{A}$ such that

$$\begin{array}{c}{\displaystyle {\int }_{\mathcal{J}}}{\phi }_l(ϵ,\tilde{z},\tilde{y})dϵ{\le }_{LU}{\displaystyle {\int }_{\mathcal{J}}}{\phi }_l(ϵ,\overline{z},\overline{y})dϵ,\forall l\in {\Gamma }_r\\ {\int }_{\mathcal{J}}{\phi }_{i_0}(ϵ,\tilde{z},\tilde{y})dϵ{<}_{LU}{\int }_{\mathcal{J}}{\phi }_{i_0}(ϵ,\overline{z},\overline{y})dϵ,\mathrm{for}\mathrm{some}{i}_0\in {\Gamma }_r.\end{array}$$

According to the definition of the order relation ${\le }_{LU}$, it follows that, for any $l\in {\Gamma }_r$, we have

$$\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ\le {\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ\le {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ.\hfill \end{array}\right.$$

(1)

According to the definition of the relation ${<}_{LU}$, it follows that, for any $l\in {\Gamma }_r$,

$$\begin{array}{c}\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ\le {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ,\end{array}\right.\\ \\ \mathrm{or}\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ\le {\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ,\hfill \end{array}\right.\\ \\ \mathrm{or}\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\tilde{z},\tilde{y})dϵ<{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ.\hfill \end{array}\right.\end{array}$$

(2)

Since $\overline{v}=\left({\overline{v}}^L,{\overline{v}}^U\right)=\left(\left({\overline{v}}_1^L,\dots ,{\overline{v}}_r^L\right),\left({\overline{v}}_1^U,\dots ,{\overline{v}}_r^U\right)\right)\ge 0$ with ${\overline{v}}^L\ge 0$ and ${\overline{v}}^U\ge 0$, Equations (1) and (2) imply that the inequality

$$\begin{array}{c}{\int }_{\mathcal{J}}\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{\phi }_l^L(ϵ,\tilde{z},\tilde{y})+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{\phi }_l^U(ϵ,\tilde{z},\tilde{y})\right\}dϵ<\\ {\int }_{\mathcal{J}}\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{\phi }_l^L(ϵ,\overline{z},\overline{y})+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{\phi }_l^U(ϵ,\overline{z},\overline{y})\right\}dϵ\end{array}$$

holds. This contradicts the assumption that $(\overline{z},\overline{y})\in \mathcal{A}$ is an optimal solution of ${(weight-P)}_{\overline{v}}$. Hence, $(\overline{z},\overline{y})\in \mathcal{A}$ is an $LU$-efficient solution of (P), which completes the proof of the theorem. □

Now, we give an example of a multiple cost interval-valued extremization problem, which we solve using the weighting method to support our result established in Theorem 2.

Example 1.

Let $\mathcal{J}=[0,1],n=s=m=1,r=2$, and consider the following multi-cost interval-valued minimization problem (P1), defined as

$$\begin{array}{c}\hfill {\displaystyle \left(\mathrm{P}1\right)\underset{(z,y)}{min}{\int }_{\mathcal{J}}\phi (ϵ,z,y)dϵ=\underset{(z,y)}{min}\left({\int }_{\mathcal{J}}{\phi }_1(ϵ,z,y)dϵ,{\int }_{\mathcal{J}}{\phi }_2(ϵ,z,y)dϵ\right)}\\ \hfill {\displaystyle =\underset{(z,y)}{min}\left(\left[{\int }_0^1(2arctanz\left(ϵ\right)+z\left(ϵ\right))dϵ,{\int }_0^1(2arctanz\left(ϵ\right)+z\left(ϵ\right)+1)dϵ\right],\right.}\\ \hfill {\displaystyle \left.\left[{\int }_0^1-e^{-z\left(ϵ\right)}dϵ,{\int }_0^1(-e^{-z\left(ϵ\right)}+z\left(ϵ\right))dϵ\right]\right)}\end{array}$$

$$\mathrm{subject}\mathrm{to}g(ϵ,z,y)=-z\left(ϵ\right)+z^2\left(ϵ\right)\leqq 0,\forall ϵ\in \mathcal{J},$$

$$h(ϵ,z,y)-\dot{z}=y\left(ϵ\right)-\dot{z}\left(ϵ\right)=0,\forall ϵ\in \mathcal{J},$$

$$z\left(0\right)=z\left(1\right)=0.$$

As it follows from the formulation of (P1), we have

$$\begin{array}{c}{\phi }_1^L(ϵ,z,y)=2arctanz\left(ϵ\right)+z\left(ϵ\right),{\phi }_1^U(ϵ,z,y)=2arctanz\left(ϵ\right)+z\left(ϵ\right)+1,\\ {\phi }_2^L(ϵ,z,y)=-e^{-z\left(ϵ\right)},{\phi }_2^U(ϵ,z,y)=-e^{-z\left(ϵ\right)}+z\left(ϵ\right).\end{array}$$

We now use the weighting method for solving (P1). Let ${\overline{v}}_1=\left({\overline{v}}_1^L,{\overline{v}}_1^U\right)=$$\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)>0$ and ${\overline{v}}_2=\left({\overline{v}}_2^L,{\overline{v}}_2^U\right)=\left({\displaystyle \frac{1}{2}},0\right)\ge 0$. The associated auxiliary weighting control problem ${\left(WP1\right)}_{\overline{v}}$ is defined by

$${\left(WP1\right)}_{\overline{v}}\underset{(z,y)}{min}\Pi (z,y)=\underset{(z,y)}{min}{\int }_{\mathcal{J}}\left\{\sum _{l=1}^2{v}_l^L{\phi }_l^L(ϵ,z,y)+\sum _{l=1}^2{v}_l^U{\phi }_l^U(ϵ,z,y)\right\}dϵ$$

$$=\underset{(z,y)}{min}{\int }_0^1\left(2arctanz\left(ϵ\right)-{\displaystyle \frac{1}{2}}{e}^{-z\left(ϵ\right)}+z\left(ϵ\right)+{\displaystyle \frac{1}{2}}\right)dϵ$$

$$\mathrm{subject}\mathrm{to}g(ϵ,z,y)=-z\left(ϵ\right)+z^2\left(ϵ\right)\leqq 0,\forall ϵ\in \mathcal{J},$$

$$h(ϵ,z,y)-\dot{z}=y\left(ϵ\right)-\dot{z}\left(ϵ\right)=0,\forall ϵ\in \mathcal{J},$$

$$z\left(0\right)=z\left(1\right)=0.$$

The set of all feasible solutions of ${\left(WP1\right)}_{\overline{v}}$ is given by

$$\mathcal{A}=\{z\left(ϵ\right)\in \mathbb{R}:z\left(0\right)=z\left(1\right)=0,0\le z\left(ϵ\right)\le 1,y\left(ϵ\right)-\dot{z}\left(ϵ\right)=0,\forall ϵ\in [0,1]\}$$

and $(\overline{z}\left(ϵ\right),\overline{y}\left(ϵ\right))=(0,0)$ is an optimal solution in ${\left(WP1\right)}_{\overline{v}}$. Further, since all the hypotheses of Theorem 2 are satisfied, $(\overline{z}\left(ϵ\right),\overline{y}\left(ϵ\right))=(0,0)$ is an $LU$-efficient solution of (P1) (see Figure 1).

Figure 1. Dynamics of the state function z(t). (a) M1 and M2, (b) M3 and M4.

Remark 1.

Based on the previous illustrative example, performing a comparative analysis of the results obtained using the weighting technique versus traditional optimization methods, we can easily conclude the effectiveness of the weighting technique in solving multiple cost optimization problems, that is, the conversion of a vector problem to a scalar one.

Now, under suitable convexity hypotheses, we prove the converse result to those established in Theorems 1 and 2.

Theorem 3.

Let each objective functional ${\int }_{\mathcal{J}}{\phi }_l(ϵ,z,y)dϵ,l\in {\Gamma }_r$, be a convex interval-valued controlled functional on the convex set $K\times Y$. If $(\overline{z},\overline{y})\in \mathcal{A}$ is a weakly $LU$-efficient solution in (P), then there exists $\overline{v}=\left({\overline{v}}^L,{\overline{v}}^U\right)\ge 0$, where ${\overline{v}}^L=\left({\overline{v}}_1^L,\dots ,{\overline{v}}_r^L\right),{\overline{v}}^U=\left({\overline{v}}_1^U,\dots ,{\overline{v}}_r^U\right)\in {\mathbb{R}}^r$, such that $(\overline{z},\overline{y})\in \mathcal{A}$ is an optimal solution of the auxiliary weighting control problem ${(weight-P)}_{\overline{v}}$.

Proof. 

Let $(\overline{z},\overline{y})\in \mathcal{A}$ be a weakly $LU$-efficient solution in (P). Then, according to Definition 2, there is no other feasible solution $(z,y)\in \mathcal{A}$ such that

$${\int }_{\mathcal{J}}{\phi }_l(ϵ,z,y)dϵ{<}_{LU}{\int }_{\mathcal{J}}{\phi }_l(ϵ,\overline{z},\overline{y})dϵ,\forall l\in {\Gamma }_r.$$

From the definition of the order relation ${<}_{LU}$, it follows that, for every $l\in {\Gamma }_r$, we have

$$\begin{array}{c}\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,z,y)dϵ<{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,z,y)dϵ\le {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ,\hfill \end{array}\right.or\left\{\begin{array}{c}{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,z,y)dϵ\le {\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,z,y)dϵ<{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ,\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^L(ϵ,z,y)dϵ<{\int }_{\mathcal{J}}{\phi }_l^L(ϵ,\overline{z},\overline{y})dϵ\hfill \\ \\ {\int }_{\mathcal{J}}{\phi }_l^U(ϵ,z,y)dϵ<{\int }_{\mathcal{J}}{\phi }_l^U(ϵ,\overline{z},\overline{y})dϵ.\hfill \end{array}\right.\end{array}$$

(3)

By assumption, each objective functional ${\int }_{\mathcal{J}}{\phi }_l(ϵ,z,y)dϵ,l\in {\Gamma }_r$, is a convex interval-valued controlled functional on $\mathcal{A}$. Then, by Definition 1, it follows that the functionals ${\int }_{\mathcal{J}}{\phi }_l^L(ϵ,z,y)dϵ$ and ${\int }_{\mathcal{J}}{\phi }_l^U(ϵ,z,y)dϵ,l\in {\Gamma }_r$, are convex on $\mathcal{A}$. Since the system of inequalities (3) has no solution for $(\overline{z},\overline{y})\in \mathcal{A}$, consequently, by Lemma 1, there exist ${\overline{v}}^L,{\overline{v}}^U\in {\mathbb{R}}^r$ with

$$\left({\overline{v}}^L,{\overline{v}}^U\right)=\left(\left({\overline{v}}_1^L,\dots ,{\overline{v}}_r^L\right),\left({\overline{v}}_1^U,\dots ,{\overline{v}}_r^U\right)\right)\ge 0$$

such that the inequality

$$\begin{array}{cc}& {\int }_{\mathcal{J}}\left\{\sum _{l=1}^r{\overline{v}}_l^L{\phi }_l^L(ϵ,z,y)+\sum _{l=1}^r{\overline{v}}_l^U{\phi }_l^U(ϵ,z,y)\right\}dϵ\ge \hfill \\ & {\int }_{\mathcal{J}}\left\{\sum _{l=1}^r{\overline{v}}_l^L{\phi }_l^L(ϵ,\overline{z},\overline{y})+\sum _{l=1}^r{\overline{v}}_l^U{\phi }_l^U(ϵ,\overline{z},\overline{y})\right\}dϵ\hfill \end{array}$$

holds for all $(z,y)\in \mathcal{A}$. This means, by Definition 4, that $(\overline{z},\overline{y})\in \mathcal{A}$ is an optimal solution of the auxiliary weighting control problem ${(weight-P)}_{\overline{v}}$, which completes the proof of this theorem 3. □

4. Conclusions

In this study, we applied a weighting technique to investigate and solve a given multi-cost interval extremization problem. Firstly, we determined the auxiliary weighting extremization problem using this method. Then, under appropriate convexity assumptions, we revealed the connection between the optimal solution of the weighting extremization problem created to solve the main problem and the (weakly) $LU$-efficient solution of the considered multi-cost interval extremization problem. The literature indicated that the weighting method is one of the techniques for solving vector optimization problems (in this case, for solving the multi-cost interval-valued extremization problems). Additionally, the methods used in this study appear to produce similar results for other types of multi-cost interval-valued extremization problems. In future research, we will explore these findings further.