On a Weighting Technique for Multiple Cost Optimization Problems with Interval Values

Abstract

This paper deals with a weighting technique for a class of multiple cost optimization problems with interval values. More specifically, we introduce a multiobjective interval-valued controlled model and investigate it by applying the weighting method. In this regard, several characterization theorems are derived. Moreover, a numerical example is formulated. Based on the provided illustrative example and 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.

1. Introduction

As is well known, optimization theory has various and multiple applications in real-world problems. Over time, several techniques and methods associated with optimization and control theories have been introduced and studied by researchers working in different scientific areas. Salehnia et al. [1], by using Multi-Level Thresholding Image Segmentation (MTIS) methods, established the threshold vector that gives the best performance of the image segmentation system. Moore [2,3,4] showed that interval analysis provides a powerful and important set of ingredients with direct applicability to various problems in scientific computing. Also, Charnes et al. [5] formulated an algorithm for solving interval-valued linear programming problems. Later, interval analysis attracted more and more researchers. Thus, Alefeld and Herzberge [6] presented an introduction to interval computations. Since interval analysis has applications in optimization theory, many optimality and duality criteria have been established by various authors. Therefore, Giannessi [7] presented alternative theorems associated with generalized systems. Ishibuchi and Tanaka [8] studied multiobjective optimization governed by interval-valued objective functions. Pereira [9] established a control design for autonomous vehicles by providing a dynamic optimization perspective. About ten years ago, Jana and Panda [10] considered a vector optimization problem in an uncertain environment, where the objective and constraint functions were interval-valued functions. Relatively recently, Ahmad et al. [11] formulated optimality conditions associated with a class of multiobjective programming problems determined by interval-valued objective functions. Moreover, Ahmad et al. [12] reported sufficiency and duality results for interval-valued variational models. Debnath and Pokharna [13] investigated optimality and duality in interval-valued variational problems with B-(p, r)-invexity assumptions. Wu [14] analyzed some interval-valued optimization problems based on the null set notion. Jha et al. [15] established a characterization of $LU$-efficiency and formulated saddle-point criteria for F-approximated multiple-objective interval-valued optimization problems. Guo et al. [16] solved a nonsmooth interval-valued extremization problem by considering interval-valued symmetric invexity hypotheses. Rani and Kummari [17] formulated duality theorems for fractional variational problems with interval values via convexificators. Treanţă and Ciontescu [18] considered a family of optimal control problems generated by generalized invariant convex interval-valued functionals. Ye [19] formulated necessary and sufficient conditions of optimality for a mathematical program with equilibrium constraints. Joshi [20] presented optimality and duality associated with a nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of the order $\sigma >0$. Su [21] established optimality and duality results for nonsmooth mathematical programming problems under equilibrium constraints.

In this paper, based on the previously mentioned research works, we apply a weighting technique for a class of multiple cost optimization problems with interval values. More specifically, we introduce a multiobjective interval-valued controlled model and investigate it by applying the weighting method. In this regard, several characterization theorems are derived. The efficiency and effectiveness of the proposed technique is demonstrated by a numerical example. The main novel elements included in this study are given by the new family of optimization problems, the concept of convexity associated with interval-valued controlled multiple integral functionals, and the innovative proofs associated with the principal results. The weighting method (for the first time applied to the considered class of optimization problems) must be understood as follows: given a vector-valued functional, having multiple integrals as components and each component is an interval-valued controlled multiple integral functional, our aim is to minimize it by using the weighting technique, namely, by minimizing a scalar multiple integral functional defined as a sum of the lower and upper ends of the considered intervals multiplied by some positive scalars (weights). Thus, since the limitations of previous studies and these gaps are addressed by this study, 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 analysis, 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 $\alpha ={\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_r\right)}^T$ and $\beta ={\left({\beta }_1,{\beta }_2,\dots ,{\beta }_r\right)}^T$ in ${\mathbb{R}}^r$, we define them as follows:

(i)

$\alpha =\beta ⟺{\alpha }_l={\beta }_l,\forall l\in {\Omega }_r$;

(ii)

$\alpha <\beta ⟺{\alpha }_l<{\beta }_l,\forall l\in {\Omega }_r$;

(iii)

$\alpha \leqq \beta ⟺{\alpha }_l\le {\beta }_l,\forall l\in {\Omega }_r$;

(iv)

$\alpha \le \beta ⟺\alpha \leqq \beta$ and $\alpha \ne \beta$,

where ${\Omega }_r=\{1,2,\dots ,r\}$ is an index set. We consider K as a compact set in ${\mathbb{R}}^w$ with $a,b\in K$ two arbitrary fixed points, and Z is a family of piecewise smooth functions $\mu :K\to {\mathbb{R}}^n$ (state variable), and $\Pi$ is the family of all piecewise continuous functions $\pi :K\to {\mathbb{R}}^s$ (control variables). Furthermore, consider $X:K\times Z\times \Pi \to {\mathbb{R}}^r,Y:K\times Z\times \Pi \to {\mathbb{R}}^m$ and $H:K\times Z\times \Pi \to {\mathbb{R}}^n$ as vector-valued functionals that possess continuous differentiability concerning each of their inputs. The functional $X=X(ϵ,\mu (ϵ),\pi (ϵ\left)\right)$ is defined for an independent variable $ϵ\in K$, with $\mu :K\to {\mathbb{R}}^n$ as an n-dimensional piecewise smooth function of $ϵ$ (${\mu }_{\gamma }\left(ϵ\right)$ denoting the partial derivative of $\mu \left(ϵ\right)$ with respect to ${ϵ}^{\gamma },\gamma =\overline{1,w}$), and $\pi :K\to {\mathbb{R}}^s$ is a s-dimensional piecewise continuous function of $ϵ$. To make the notation less complex, we will denote $\mu \left(ϵ\right),\pi \left(ϵ\right)$ and ${\mu }_{\gamma }\left(ϵ\right)$ as $\mu ,\pi$ and ${\mu }_{\gamma }$, respectively. If $X^l,l=1,\dots ,r$ are the components of the above-mentioned vector-valued function X, the partial derivatives of $X^l$ with respect to $ϵ, \mu$, and $\pi$ are denoted as $X_{ϵ}^l, X_{\mu }^l$, and $X_{\pi }^l$, respectively. More precisely, $X_{\mu }^l$ and $X_{\pi }^l$ are defined as the vectors ${\left({\displaystyle \frac{\partial X^l}{\partial {\mu }_1}},\dots ,{\displaystyle \frac{\partial X^l}{\partial {\mu }_n}}\right)}^T$ and ${\left({\displaystyle \frac{\partial X^l}{\partial {\pi }_1}},\dots ,{\displaystyle \frac{\partial X^l}{\partial {\pi }_s}}\right)}^T$, respectively. Similarly, the first-order partial derivatives $g_{\mu },h_{\mu }$ and $g_{\pi },h_{\pi }$ of the vector-valued functionals Y and H, respectively, can be expressed using matrices with $m\left(n\right)$ rows instead of a single row.

Consider the family $\mathcal{J}\left(\mathbb{R}\right)$ 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 $\mathcal{R}=\left[{\underline{r}}^L,{\overline{r}}^U\right]\in \mathcal{J}\left(\mathbb{R}\right)$, 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 $\mathcal{J}\left(\mathbb{R}\right)$. 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 [2,3], Moore et al.[4], and Alefeld and Herzberger [6]) 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 $\mathcal{J}\left(\mathbb{R}\right)$, 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 $\mathcal{J}\left(\mathbb{R}\right)$ 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 by $\mathcal{J}\left({\mathbb{R}}^p\right)$ the family of all vectors of compact real intervals. For two vectors of 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 $l\in {\Omega }_p$. Similarly, we will use $\mathcal{R}{<}_{LU}\mathcal{P}$ to indicate that ${\mathcal{R}}_l{\le }_{LU}{\mathcal{P}}_l$, for all $l\in {\Omega }_p$, and ${\mathcal{R}}_{l^{*}}{<}_{LU}{\mathcal{P}}_{l^{*}}$, for at least one $l^{*}\in {\Omega }_p$.

Now, we will formulate the convexity concept associated with an interval-valued controlled multiple integral functional. In particular, we use the very straightforward concept of convexity introduced by Wu [22] and Treanţă [23,24].

Definition 1.

Let $\Phi :Z\times \Pi \to \mathcal{J}\left(\mathbb{R}\right)$ be defined by

$${\displaystyle \Phi (\mu ,\pi )={\int }_{K}X(ϵ,\mu ,{\mu }_{\gamma },\pi )dϵ=\left[{\int }_K{X}^L(ϵ,\mu ,{\mu }_{\gamma },\pi )dϵ,{\int }_K{X}^U(ϵ,\mu ,{\mu }_{\gamma },\pi )dϵ\right],}$$

with $X:K\times Z\times Z\times \Pi \to \mathcal{J}\left(\mathbb{R}\right)$ and $X^L,X^U:K\times Z\times Z\times \Pi \to \mathbb{R}$ continuously differentiable functionals. Then, ${\displaystyle \Phi (\mu ,\pi )={\int }_{K}X(ϵ,\mu ,{\mu }_{\gamma },\pi )dϵ}$ is said to be a convex interval-valued controlled multiple integral functional on $Z\times \Pi$ if the inequalities

$$\begin{array}{c}{\int }_K{X}^L(ϵ,\mu ,{\mu }_{\gamma },\pi )dϵ-{\int }_K{X}^L(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })dϵ\ge \\ {\int }_K\{{(\mu -\overline{\mu })}^T{X}_{\mu }^L(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })+{({\mu }_{\gamma }-{\overline{\mu }}_{\gamma })}^T{X}_{{\mu }_{\gamma }}^L(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })\\ +{(\pi -\overline{\pi })}^T{X}_{\pi }^L(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })\}dϵ,\\ {\int }_K{X}^U(ϵ,\mu ,{\mu }_{\gamma },\pi )dϵ-{\int }_K{X}^U(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })dϵ\ge \\ {\int }_K\{{(\mu -\overline{\mu })}^T{X}_{\mu }^U(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })+{({\mu }_{\gamma }-{\overline{\mu }}_{\gamma })}^T{X}_{{\mu }_{\gamma }}^U(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })\\ +{(\pi -\overline{\pi })}^T{X}_{\pi }^U(ϵ,\overline{\mu },{\overline{\mu }}_{\gamma },\overline{\pi })\}dϵ\end{array}$$

hold for all $\mu ,\overline{\mu }\in Z,\pi ,\overline{\pi }\in \Pi$.

The alternative lemma presented below is a specific case of the more general results established in Giannessi [7] and Giorgi [25] for convex vector optimization problems.

Lemma 1.

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

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

has no solution, then there exists $\delta =\left({\delta }^L,{\delta }^U\right)\ge 0$, 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

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

3. Main Results

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

$$\begin{array}{c}\left(\mathrm{P}\right){\displaystyle \underset{(\mu ,\pi )}{min}}{\int }_{K}X(ϵ,\mu ,\pi )dϵ=\underset{(\mu ,\pi )}{min}\left({\int }_K{X}_1(ϵ,\mu ,\pi )dϵ,\dots ,{\int }_K{X}_r(ϵ,\mu ,\pi )dϵ\right)\\ \mathrm{subject}\mathrm{to}Y(ϵ,\mu ,\pi )\leqq 0,\forall ϵ\in K,\\ \left(H_{\gamma }(ϵ,\mu ,\pi )\right)=\left({\mu }_{\gamma }\right),\gamma =\overline{1,w},\forall ϵ\in K,\\ \mu \left(a\right)=a_0,\mu \left(b\right)=b_0,\end{array}$$

where ${\displaystyle {\int }_K{X}_l(ϵ,\mu ,\pi )dϵ=\left[{\int }_K{X}_l^L(ϵ,\mu ,\pi )dϵ,{\int }_K{X}_l^U(ϵ,\mu ,\pi )dϵ\right]}$ for each $l\in {\Omega }_r$, $a_0,b_0\in {\mathbb{R}}^n$ are given, and $X_l^L,X_l^U:I\times Z\times \Pi \to \mathbb{R},l\in {\Omega }_r,Y:I\times Z\times \Pi \to {\mathbb{R}}^m$ and $H:I\times Z\times \Pi \to {\mathbb{R}}^w$ are $C^1$-class functionals. Let

$$F=\{(\mu ,\pi )\in Z\times \Pi :Y(ϵ,\mu ,\pi )\leqq 0,\left(H_{\gamma }(ϵ,\mu ,\pi )\right)=\left({\mu }_{\gamma }\right),\gamma =\overline{1,w},\forall ϵ\in K,$$

$$\mu \left(a\right)=a_0,\mu \left(b\right)=b_0\}$$

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 points, as below.

Definition 2.

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

$${\int }_K{X}_l(ϵ,\mu ,\pi )dϵ{<}_{LU}{\int }_K{X}_l(ϵ,\overline{\mu },\overline{\pi })dϵ,\forall l\in {\Omega }_r.$$

Definition 3.

A feasible solution $(\overline{\mu },\overline{\pi })\in F$ is said to be an $LU$-efficient point of (P) if and only if there exists no other $(\mu ,\pi )\in F$, such that

$$\begin{array}{c}{\int }_K{X}_l(ϵ,\mu ,\pi )dϵ{\le }_{LU}{\int }_K{X}_l(ϵ,\overline{\mu },\overline{\pi })dϵ,\forall l\in {\Omega }_r\\ {\int }_K{X}_{i_0}(ϵ,\mu ,\pi )dϵ{<}_{LU}{\int }_K{X}_{i_0}(ϵ,\overline{\mu },\overline{\pi })dϵ,\mathit{for}\mathit{some}{i}_0\in {\Omega }_r.\end{array}$$

In this section, to investigate a weakly $LU$-efficient point and/or an $LU$-efficient point of (P), we use the weighting approach (see Antczak [26]). Thus, for this purpose, an auxiliary weighting control problem is introduced for the considered multiple cost interval-valued extremization problem as follows:

$${\left(\mathit{\text{weight-P}}\right)}_v\underset{(\mu ,\pi )}{min}\Gamma (\mu ,\pi )=\underset{(\mu ,\pi )}{min}{\int }_K\left\{\sum _{l=1}^r{v}_l^L{X}_l^L(ϵ,\mu ,\pi )+\sum _{l=1}^r{v}_l^U{X}_l^U(ϵ,\mu ,\pi )\right\}dϵ$$

$$\begin{array}{c}\mathrm{subject}\mathrm{to}Y(ϵ,\mu ,\pi )\leqq 0,\forall ϵ\in K,\\ \left(H_{\gamma }(ϵ,\mu ,\pi )\right)=\left({\mu }_{\gamma }\right),\gamma =\overline{1,w},\forall ϵ\in K,\\ \mu \left(a\right)=a_0,\mu \left(b\right)=b_0,\end{array}$$

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 $(\overline{\mu },\overline{\pi })\in F$ is said to be an optimal solution of ${\left(weight\text{-}P\right)}_v$ if the inequality

$$\begin{array}{cc}& {\int }_K\left\{\sum _{l=1}^r{v}_l^L{X}_l^L(ϵ,\overline{\mu },\overline{\pi })+\sum _{l=1}^r{\overline{v}}_l^U{X}_l^U(ϵ,\overline{\mu },\overline{\pi })\right\}dϵ\hfill \\ & \le {\int }_K\left\{\sum _{l=1}^r{v}_l^L{X}_l^L(ϵ,\mu ,\pi )+\sum _{l=1}^r{v}_l^U{X}_l^U(ϵ,\mu ,\pi )\right\}dϵ\hfill \end{array}$$

holds for all $(\mu ,\pi )\in F$.

Theorem 1.

Let $(\overline{\mu },\overline{\pi })\in F$ be an optimal solution of ${\left(weight\text{-}P\right)}_{\overline{v}}$. 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 $i_0\in {\Omega }_r$. Then $(\overline{\mu },\overline{\pi })\in F$ is a weakly $LU$-efficient point of (P).

Proof. 

By assumption, $(\overline{\mu },\overline{\pi })\in F$ is an optimal solution of ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$. We assume, on the contrary, that $(\overline{\mu },\overline{\pi })\in F$ is not a weakly $LU$-efficient point to (P). Therefore, according to Definition 2, it follows that there exists another $(\tilde{\mu },\tilde{\pi })\in F$, such that

$${\int }_K{X}_l(ϵ,\tilde{\mu },\tilde{\pi })dϵ{<}_{LU}{\int }_K{X}_l(ϵ,\overline{\mu },\overline{\pi })dϵ,\forall l\in {\Omega }_r.$$

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

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ\le {\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ,\hfill \end{array}\right.\\ \\ \hfill \mathrm{or}\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ\le {\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ,\hfill \end{array}\right.\\ \\ \hfill \mathrm{or}\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ.\hfill \end{array}\right.\end{array}$$

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 $i_0\in {\Omega }_r$, the above system of inequalities yields that the inequality

$$\begin{array}{c}{\int }_K\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })\right\}dϵ<\\ {\int }_K\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{X}_l^L(ϵ,\overline{\mu },\overline{\pi })+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{X}_l^U(ϵ,\overline{\mu },\overline{\pi })\right\}dϵ\end{array}$$

holds. This contradicts the assumption that $(\overline{\mu },\overline{\pi })\in F$ is an optimal solution of ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$. Hence, $(\overline{\mu },\overline{\pi })\in F$ is a weakly $LU$-efficient point of the considered control problem (P), which completes the proof of the theorem. □

Theorem 2.

Let $(\overline{\mu },\overline{\pi })\in F$ be an optimal solution of ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$. 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 $(\overline{\mu },\overline{\pi })\in F$ is an $LU$-efficient point of (P).

Proof. 

By assumption, $(\overline{\mu },\overline{\pi })\in F$ is an optimal solution of ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$. We assume, contrary to the result, that $(\overline{\mu },\overline{\pi })\in F$ is not an $LU$-efficient point to (P). Therefore, according to Definition 3, it follows that there exists another $(\tilde{\mu },\tilde{\pi })\in F$, such that

$$\begin{array}{c}{\int }_K{X}_l(ϵ,\tilde{\mu },\tilde{\pi })dϵ{\le }_{LU}{\int }_K{X}_l(ϵ,\overline{\mu },\overline{\pi })dϵ,\forall l\in {\Omega }_r\\ {\int }_K{X}_{i_0}(ϵ,\tilde{\mu },\tilde{\pi })dϵ{<}_{LU}{\int }_K{X}_{i_0}(ϵ,\overline{\mu },\overline{\pi })dϵ,\mathrm{for}\mathrm{some}{i}_0\in {\Omega }_r.\end{array}$$

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

$$\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ\le {\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ\le {\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ.\hfill \end{array}\right.$$

(1)

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

$$\begin{array}{c}\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ\le {\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ,\end{array}\right.\\ \\ \mathrm{or}\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ\le {\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ,\hfill \end{array}\right.\\ \\ \mathrm{or}\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })dϵ<{\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ.\hfill \end{array}\right.\end{array}$$

(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 ${\overline{v}}^U\ge 0$, (1) and (2) imply that the inequality

$$\begin{array}{c}{\int }_K\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{X}_l^L(ϵ,\tilde{\mu },\tilde{\pi })+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{X}_l^U(ϵ,\tilde{\mu },\tilde{\pi })\right\}dϵ<\\ {\int }_K\left\{{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^L{X}_l^L(ϵ,\overline{\mu },\overline{\pi })+{\displaystyle \sum _{l=1}^r}{\overline{v}}_l^U{X}_l^U(ϵ,\overline{\mu },\overline{\pi })\right\}dϵ\end{array}$$

holds. This contradicts the assumption that $(\overline{\mu },\overline{\pi })\in F$ is an optimal solution of ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$. Hence, $(\overline{\mu },\overline{\pi })\in F$ is an $LU$-efficient point 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 $K=[0,1]\times [0,1]={[0,1]}^2,n=s=m=1,r=w=2$, and consider the following multiple cost interval-valued minimization problem (P1), defined as

$$\begin{array}{c}\hfill {\displaystyle \left(\mathrm{P}1\right)\underset{(\mu ,\pi )}{min}{\int }_{K}X(ϵ,\mu ,\pi )dϵ=\underset{(\mu ,\pi )}{min}\left({\int }_K{X}_1(ϵ,\mu ,\pi )dϵ,{\int }_K{X}_2(ϵ,\mu ,\pi )dϵ\right)}\\ \hfill {\displaystyle =\underset{(\mu ,\pi )}{min}\left(\left[{\int }_{{[0,1]}^2}(2arcsin\mu \left(ϵ\right)+\mu \left(ϵ\right))dϵ,{\int }_{{[0,1]}^2}(2arcsin\mu \left(ϵ\right)+\mu \left(ϵ\right)+1)dϵ\right],\right.}\\ \hfill {\displaystyle \left.\left[{\int }_{{[0,1]}^2}-e^{-\mu \left(ϵ\right)}dϵ,{\int }_{{[0,1]}^2}(-e^{-\mu \left(ϵ\right)}+\mu \left(ϵ\right))dϵ\right]\right)}\end{array}$$

$$\mathit{subject}\mathit{to}-\mu \left(ϵ\right)+{\mu }^2\left(ϵ\right)\le 0,\forall ϵ\in K,$$

$$\pi \left(ϵ\right)-{\mu }_{\gamma }\left(ϵ\right)=0,\gamma =1,2,\forall ϵ\in K,$$

$$\mu (0,0)=\mu (1,1)=0.$$

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

$$\begin{array}{c}{X}_1^L(ϵ,\mu ,\pi )=2arcsin\mu \left(ϵ\right)+\mu \left(ϵ\right),X_1^U(ϵ,\mu ,\pi )=2arcsin\mu \left(ϵ\right)+\mu \left(ϵ\right)+1,\\ X_2^L(ϵ,\mu ,\pi )=-e^{-\mu \left(ϵ\right)},X_2^U(ϵ,\mu ,\pi )=-e^{-\mu \left(ϵ\right)}+\mu \left(ϵ\right).\end{array}$$

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

$${\left(WP1\right)}_{\overline{v}}\underset{(\mu ,\pi )}{min}\Gamma (\mu ,\pi )=\underset{(\mu ,\pi )}{min}{\int }_K\left\{\sum _{l=1}^2{v}_l^L{X}_l^L(ϵ,\mu ,\pi )+\sum _{l=1}^2{v}_l^U{X}_l^U(ϵ,\mu ,\pi )\right\}dϵ$$

$$=\underset{(\mu ,\pi )}{min}{\int }_{{[0,1]}^2}\left(2arcsin\mu \left(ϵ\right)-{\displaystyle \frac{1}{2}}{e}^{-\mu \left(ϵ\right)}+\mu \left(ϵ\right)+{\displaystyle \frac{1}{2}}\right)dϵ$$

$$\mathit{subject}\mathit{to}-\mu \left(ϵ\right)+{\mu }^2\left(ϵ\right)\leqq 0,\forall ϵ\in K,$$

$$\pi \left(ϵ\right)-{\mu }_{\gamma }\left(ϵ\right)=0,\gamma =1,2,\forall ϵ\in K,$$

$$\mu (0,0)=\mu (1,1)=0.$$

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

$$F=\left\{\mu \right(ϵ)\in \mathbb{R}:\mu (0,0)=\mu (1,1)=0,0\le \mu (ϵ)\le 1,$$

$$\pi \left(ϵ\right)-{\mu }_{\gamma }\left(ϵ\right)=0,\gamma =1,2,\forall ϵ\in {[0,1]}^2\}$$

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

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

Theorem 3.

Let each objective multiple integral functional

$${\displaystyle {\int }_K{X}_l(ϵ,\mu ,\pi )dϵ,l\in {\Omega }_r}$$

be a convex interval-valued controlled multiple integral functional on the convex set $Z\times \Pi$. If $(\overline{\mu },\overline{\pi })\in F$ is a weakly $LU$-efficient point 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 $(\overline{\mu },\overline{\pi })\in F$ is an optimal solution of the auxiliary weighting control problem ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$.

Proof. 

Let $(\overline{\mu },\overline{\pi })\in F$ be a weakly $LU$-efficient point in (P). Then, according to Definition 2, there is no other feasible solution $(\mu ,\pi )\in F$ such that

$${\int }_K{X}_l(ϵ,\mu ,\pi )dϵ{<}_{LU}{\int }_K{X}_l(ϵ,\overline{\mu },\overline{\pi })dϵ,\forall l\in {\Omega }_r.$$

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

$$\begin{array}{c}\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\mu ,\pi )dϵ<{\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\mu ,\pi )dϵ\le {\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ,\hfill \end{array}\right.or\left\{\begin{array}{c}{\int }_K{X}_l^L(ϵ,\mu ,\pi )dϵ\le {\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\mu ,\pi )dϵ<{\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ,\hfill \\ \\ {\int }_K{X}_l^L(ϵ,\mu ,\pi )dϵ<{\int }_K{X}_l^L(ϵ,\overline{\mu },\overline{\pi })dϵ\hfill \\ \\ {\int }_K{X}_l^U(ϵ,\mu ,\pi )dϵ<{\int }_K{X}_l^U(ϵ,\overline{\mu },\overline{\pi })dϵ.\hfill \end{array}\right.\end{array}$$

(3)

By assumption, each objective multiple integral functional ${\int }_K{X}_l(ϵ,\mu ,\pi )dϵ,l\in {\Omega }_r$ is a convex interval-valued controlled multiple integral functional on F. Then, according to Definition 1, it follows that the multiple integral functionals ${\int }_K{X}_l^L(ϵ,\mu ,\pi )dϵ$ and ${\int }_K{X}_l^U(ϵ,\mu ,\pi )dϵ,l\in {\Omega }_r$, are convex on F. Since the system of inequalities (3) has no solution for $(\overline{\mu },\overline{\pi })\in F$, 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

$$\begin{array}{cc}& {\int }_K\left\{\sum _{l=1}^r{\overline{v}}_l^L{X}_l^L(ϵ,\mu ,\pi )+\sum _{l=1}^r{\overline{v}}_l^U{X}_l^U(ϵ,\mu ,\pi )\right\}dϵ\ge \hfill \\ & {\int }_K\left\{\sum _{l=1}^r{\overline{v}}_l^L{X}_l^L(ϵ,\overline{\mu },\overline{\pi })+\sum _{l=1}^r{\overline{v}}_l^U{X}_l^U(ϵ,\overline{\mu },\overline{\pi })\right\}dϵ\hfill \end{array}$$

holds for all $(\mu ,\pi )\in F$. This means, according to Definition 4, that $(\overline{\mu },\overline{\pi })\in F$ is an optimal solution of the auxiliary weighting control problem ${\left(\mathit{\text{weight-P}}\right)}_{\overline{v}}$, which completes the proof of this theorem. □

4. Discussion of Technical and Distinguished Issues

In this study, we introduced a multiobjective interval-valued controlled model and investigated it by applying the weighting method. In this regard, various characterization theorems have been formulated and proved. The efficiency and effectiveness of the proposed technique has been illustrated by a numerical example. Based on the previous illustrative example, performing a comparative analysis of the results obtained using the weighting technique versus traditional optimization methods, we can easy 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. The weighting method (for the first time applied to the considered class of optimization problems) must be understood as follows: given a vector-valued functional, having multiple integrals as components and each component being an interval-valued controlled multiple integral functional, our aim is to minimize it by using the weighting technique, namely, by minimizing a scalar multiple integral functional defined as a sum of the lower and upper ends of the considered intervals multiplied by some positive scalars (weights).

5. Conclusions and Future Research Directions

In this paper, the authors applied a weighting technique for a class of multiple cost optimization problems with interval values. More specifically, the authors introduced a multiobjective interval-valued controlled model and solved it by using the weighting method. In this regard, several characterization theorems have been established. Moreover, to illustrate the theoretical developments, an illustrative example was formulated.

As potential extensions of the proposed weighting technique for different types of optimization problems or domains, we should mention the study of well-posedness associated with similar classes of extremization problems governed by path-independent curvilinear integral functionals. This is a specific research question or unresolved issue that could be addressed in future studies to build upon the current findings. Consequently, we note the applicability of the weighting technique to larger or more complex optimization problems.