The coefficients of objective functions and constraint functions in optimization problems are usually taken to be real numbers. In this case, the problems are categorized as deterministic optimization problems. When uncertainties are taken into account in optimization problems, the coefficients will be taken to be uncertain quantities. There are two kinds of uncertainties that can be considered, which are randomness and fuzziness.

When the randomness is taken into account, the coefficients of objective functions and constraint functions in optimization problems can be assumed to be random variables with known probability distributions. In this case, the so-called stochastic optimization problems can be studied using the probability theory. We can refer to the books written by Birge and Louveaux [1], Kall [2], Prékopa [3], Stancu-Minasian [4], and Vajda [5], which address the main stream of this topic and provide many useful techniques to solve the stochastic optimization problems.

When the fuzziness is taken into account, the coefficients of objective functions and constraint functions in optimization problems can be assumed to be fuzzy quantities. In this case, the so-called fuzzy optimization problems can be studied using the fuzzy set theory. We can refer to the books written by Słowiński [6] and Delgado et al. [7], which provide many interesting concepts and topics. On the other hand, the book edited by Słowiński and Teghem [8] presents the fusion of randomness and fuzziness in optimization problems. In particular, Inuiguchi and Ramík [9] provide a brief review of fuzzy optimization and a comparison with stochastic optimization in portfolio selection problem.

Without considering randomness and fuzziness, the bounded closed intervals in $\mathbb{R}$ can also be used to formulate the uncertainties. When the coefficients of objective functions and constraint functions in optimization problems are assumed to be bounded closed intervals in $\mathbb{R}$, the so-called interval-valued optimization problems can be studied using interval analysis by referring to Moore [10,11]. The probability distributions in stochastic optimization problems and the membership functions in fuzzy optimization problems are frequently determined by the decision-makers, which can be subjective and sometimes very difficult to determine the suitable probability distributions and the membership functions. In this case, we can consider the bounded closed intervals in optimization problems to take care of uncertainties. Although the specification of bounded closed intervals may still be judged as a subjective viewpoint, we may argue that the bounds of uncertain data, by determining the bounded closed intervals to restrict the possible observed data, are easier to be handled than specifying the probability distributions in stochastic optimization problems and membership functions in fuzzy optimization problems.

Suppose that a company produces p products and has n objectives that should be minimized. The amounts of these p products are denoted by $x_1,\cdots ,x_p$. For the ith objective, the cost for producing $x_1,\cdots ,x_p$ needs where the quantities $c_{ij}$ for $i=1\cdots ,n$ and $j=1,\cdots ,p$ are real numbers representing the demands for producing one item of these p products. The purpose of this company is to minimize these n objectives. However, owing to some unexpected situation, the exact quantities of $c_{ij}$ cannot be determined:

However, determining the suitable probability distribution functions and membership functions are not the easy tasks. Also, their mathematical forms may be complicated such that the numerical simulation is difficult to perform. The easier way is to consider the bounded closed intervals. In practical situation, owing to the fluctuation, the decision-makers can just know that $c_{ij}$ is located in the bounded closed interval $[a_{ij}^L,a_{ij}^U]$ for $i=1,\cdots ,n$ and $j=1,\cdots ,p$. In this case, this company needs to minimize the following objective functions which are the functions with interval-valued coefficients.

Ishibuchi and Tanaka [12] proposed three ordering relations on the space of all bounded and closed intervals in $\mathbb{R}$ to study the interval-valued optimization problems. Jiang et al. [13] used the ordering relation proposed by Ishibuchi and Tanaka [12], and the concept of possibility degree to transform the interval-valued optimization problems into a bi-objective optimization problem. Chanas and Kuchta [14] extended those ordering relations by considering the $t_0$ and $t_1$ cuts of intervals in $\mathbb{R}$, and also used them to propose the solution concepts of interval-valued optimization problems.

Costa et al. [15] presented a preference ordering relations on the space of all bounded and closed intervals in $\mathbb{R}$ such that many existing relations presented in the literature can be considered as the particular cases. This family of preference order relations was also used to provide the solution concepts of interval-valued optimization problems.

The Karush–Kuhn–Tucker optimality conditions in interval-valued optimization problems was studied by Wu [16], in which the Hukuhara derivative of interval-valued functions was considered. The generalization has also been performed in two different directions. Chalco-Canoet et al. [17] studied the Karush–Kuhn–Tuck optimality conditions in interval-valued optimization problems by considering the generalized Hukuhara derivative of interval-valued functions. Jayswal et al. [18] studied the Karush–Kuhn–Tucker optimality conditions in interval-valued optimization problems by considering the generalized convexity (also called invexity). Osuna-Gomez et al. [19] also studied the necessary and sufficient optimality conditions for unconstrained interval-valued optimization problems.

Li and Tian [20,21] studied the interval-valued quadratic programming problems in which the coefficients were taken to be the bounded closed intervals. The solution concept of this kind of interval-valued optimization problems was not considered. They just designed a numerical method to solve the upper bound and lower bound of uncertain objective values of the uncertain quadratic programming problems, in which the uncertainties were assumed to take all possible values from the corresponding bounded closed intervals. In other words, a lot of counterparts of the quadratic programming problem were considered such that the coefficients were taken from the bounded closed intervals. The purpose of their approach was to find the lower bound and upper bound of the objective values of all possible counterparts.

Soyster [22,23,24], Thuente [25] and Falk [26] provided some properties for inexact linear programming problems, which are a kind of interval-valued optimization problem. However, Pomerol [27] pointed out some drawbacks of Soyster’s results and also provided some mild conditions to improve Soyster’s results. The main difference between the interval-valued optimization problems and inexact programming problems is the solutions concepts imposed upon the objective functions. The solution concept in inexact programming problem uses the conventional solution concept However, the solution concept of interval-valued optimization problems follows from the solution concept of multiobjective optimization problems.

In this paper, we consider a different solution concept by introducing an equivalence relation to divide the set of all bounded closed intervals into equivalence classes such that we can consider the concept of the convex cone in the family of all bounded closed intervals in $\mathbb{R}$. The purpose is to consider the solution concept of interval-valued optimization problems using ordering cones since the ordering cone can induce a partial ordering. In this case, the solution concepts of interval-valued optimization problems can be elicited by using the similar concept of the Pareto optimal solution in multiobjective optimization problems.

In Section 2, an equivalence relation is introduced to divide the collection of all bounded closed intervals in $\mathbb{R}$ into equivalence classes. After that, we can introduce the vector structure to the family of equivalence classes such that it can turn into a vector space. In this case, we can use the technique in vector optimization to study the interval-valued multiobjective optimization problems. In Section 3, we introduce the optimality notions in the quotient set using the ordering cones, where the quotient set consists of all equivalence classes. In Section 4, the interval-valued multiobjective optimization problems can be formulated by using the ordering cones such that the solution concepts of interval-valued multiobjective optimization problems can be reasonable realized. On the other hand, the sufficient conditions to obtain the Pareto optimal solutions are also provided. In Section 5, we take into account the multiobjective optimization problem, in which the coefficients of objective functions and constraint functions are taken to be the bounded closed intervals in $\mathbb{R}$. The purpose is to show that the optimal solutions of its transformed (conventional) optimization problem are the Pareto optimal solutions of the original multiobjective optimization problem with interval-valued coefficients. In this case, in order to solve the interval-valued multiobjective optimization problems, we can simply solve their corresponding transformed (conventional) optimization problem by using the well-known technique. In Section 6, we present some practical problems. We use the results obtained in Section 4 and Section 5 to interpret the ordering cone as a partial ordering, and provide some examples to clarify the notion of ordering cones.

The bounded closed interval in $\mathbb{R}$ is also called a compact interval. Each real number $a\in \mathbb{R}$ can also be treated as a compact interval $A=[a,a]$, which can be called the degenerated interval. Let $\mathcal{I}$ denote the family of all compact intervals. Given a compact interval A, we also write $A=[a^L,a^U]$. Given two compact intervals the addition is defined by Given any $k\in \mathbb{R}$ and $A\in \mathcal{I}$, the scalar multiplication is defined by It is clear to see and

Given any compact interval $A=[a^L,a^U]$, for convenience, we also write In this case, it is clear to see

Each compact interval $A\in \mathcal{I}$ cannot have an additive inverse element. It says that the collection of all compact intervals $\mathcal{I}$ cannot form a vector space under the vector addition $A\oplus B$ and scalar multiplication $kA$ defined above. Therefore, the existing technique of vector optimization is not valid to study the interval-valued optimization problems. In order to conquer this difficulty, we introduce a scalar function $\eta :\mathcal{I}\to \mathbb{R}$ to scalarize $\mathcal{I}$ by assigning a compact interval $A\in \mathcal{I}$ to a real number $\eta \left(A\right)\in \mathbb{R}$.

Using the scalar function $\eta$, we can define a binary relation as follows. Let “∼” be a binary relation on $\mathcal{I}$ defined by

It is clear to see that the binary relation “∼” is an equivalence relation, which means that this binary relation is reflexive, symmetric, and transitive. In this case, this equivalence relation can induce a quotient set given by where is an equivalence class. We also adopt the notation ${\mathcal{I}}_{\eta }=\mathcal{I}/\sim$ to say that the family ${\mathcal{I}}_{\eta }$ of equivalence classes depends on the scalar function $\eta$.

The addition in ${\mathcal{I}}_{\eta }$ is defined by

We need to claim that the definition in (2) is well defined. In other words, given any $\overline{A}\in \langle A\rangle$ and $\overline{B}\in \langle B\rangle$, we need to show $\langle A\oplus B\rangle =\langle \overline{A}\oplus \overline{B}\rangle$. Now, we have $\eta \left(\overline{A}\right)=\eta \left(A\right)$ and $\eta \left(\overline{B}\right)=\eta \left(B\right)$. According to Definition 1, we have which shows $\overline{A}\oplus \overline{B}\sim A\oplus B$. Therefore, we obtain $\langle A\oplus B\rangle =\langle \overline{A}\oplus \overline{B}\rangle$. This says that the definition in (2) is well defined.

For convenience, we write

Then, we have which says that $\langle 0\rangle$ is a two-sided zero element of ${\mathcal{I}}_{\eta }$.

Given any compact interval A, we cannot say that $-A$ is the inverse element of A since $A\oplus (-A)$ is not a zero element of $\mathcal{I}$. This is the reason why $\mathcal{I}$ cannot form a vector space. However, we can show that $\langle -A\rangle$ is the inverse element of $\langle A\rangle$; that is to say, we can claim $\langle A\rangle +\langle -A\rangle =\langle 0\rangle$. Now, we have and by Definition 1. This also says $A\oplus (-A)\in \langle 0\rangle$. Therefore, we obtain which shows that $\langle -A\rangle$ is the inverse element of $\langle A\rangle$. In other words, we have $-\langle A\rangle =\langle -A\rangle$.

Given any $\lambda \in \mathbb{R}$ and $A\in \mathcal{I}$, the scalar multiplication in ${\mathcal{I}}_{\eta }$ is defined by

We also need to claim that the definition in (3) is well defined. In other words, given any $\overline{A}\in \langle A\rangle$, we want to show $\langle \lambda A\rangle =\langle \lambda \overline{A}\rangle$. Now, we have $\eta \left(\overline{A}\right)=\eta \left(A\right)$. According to Definition 1, we also have which says $\lambda \overline{A}\sim \lambda A$. Therefore, we obtain $\langle \lambda A\rangle =\langle \lambda \overline{A}\rangle$. This shows that the definition in (3) is well-defined. Then, we have the following proposition.

Now, we consider the product space It is clear to see that where $\langle A_i\rangle \in {\mathcal{I}}_{\eta }$ for $i=1,2,\cdots ,n$. The vector addition in ${\mathcal{I}}_{\eta }^n$ is defined by

Given any $\lambda \in \mathbb{R}$, the scalar multiplication in ${\mathcal{I}}_{\eta }^n$ is defined by

We write

It is clear to see that $\langle \mathbf{0}\rangle$ is a two-sided zero element of the product space ${\mathcal{I}}_{\eta }^n$. Given any $\langle \mathbf{A}\rangle \in {\mathcal{I}}_{\eta }$, we are going to claim that the inverse elements of $\langle \mathbf{A}\rangle$ is $\langle -\mathbf{A}\rangle$. Now, we have which says that $\langle -\mathbf{A}\rangle$ is an inverse element of $\langle \mathbf{A}\rangle$. In other words, we have $-\langle \mathbf{A}\rangle =\langle -\mathbf{A}\rangle$. The following proposition is obvious.

Each binary relation “⪯” on the vector space ${\mathcal{I}}_{\eta }^n$ is called a partial ordering on ${\mathcal{I}}_{\eta }^n$ when the following conditions are satisfied.

Suppose that “⪯” is a partial ordering on ${\mathcal{I}}_{\eta }^n$. We can show that the following set is a convex cone in the vector space ${\mathcal{I}}_{\eta }^n$. Conversely, let $\mathcal{C}$ be a convex cone in ${\mathcal{I}}_{\eta }^n$. Then, we can induce a binary relation “⪯” defined by We can show that this binary relation “⪯” is a partial ordering on ${\mathcal{I}}_{\eta }^n$. We also have

A convex cone $\mathcal{C}$ that defines a partial ordering as described above in the vector space ${\mathcal{I}}_{\eta }^n$ is called an ordering cone. By referring to Jahn [28], we can consider the optimality notions in ${\mathcal{I}}_{\eta }^n$ based on the convex cone $\mathcal{C}$.

Let $\mathcal{S}$ be a subset of the vector space ${\mathcal{I}}_{\eta }^n$, and let $\mathcal{C}$ be an ordering cone in ${\mathcal{I}}_{\eta }^n$. Then, we can obtain a corresponding partial ordering “⪯” on ${\mathcal{I}}_{\eta }^n$, which is induced from $\mathcal{C}$. More precisely, we have where $\langle \mathbf{B}\rangle -\langle \mathbf{A}\rangle \in \mathcal{C}$ means $\langle \mathbf{B}\rangle -\langle \mathbf{A}\rangle =\langle \mathbf{C}\rangle$ for some $\langle \mathbf{C}\rangle \in \mathcal{C}$. By adding $\langle \mathbf{A}\rangle$ and $-\langle \mathbf{C}\rangle$ on both sides, we obtain $\langle \mathbf{A}\rangle =\langle \mathbf{B}\rangle +(-\langle \mathbf{C}\rangle )$. It shows where

Therefore, by referring to (7), we see that $\langle {\mathbf{A}}^{*}\rangle ⪯\langle \mathbf{A}\rangle$ means

On the other hand, we see that $\langle \mathbf{A}\rangle ⪯\langle {\mathbf{A}}^{*}\rangle$ for $\langle \mathbf{A}\rangle \in \mathcal{S}$ means

Using (9) and (8), Definition 2 says that $\langle {\mathbf{A}}^{*}\rangle \in \mathcal{S}$ is a minimal element of the set $\mathcal{S}$ when the following inclusion is satisfied:

Therefore, without considering the partial ordering “⪯”, using an ordering cone $\mathcal{C}$ in the vector space ${\mathcal{I}}_{\eta }^n$, we propose the concepts of cone-extreme elements as follows.

The ordering cone $\mathcal{C}$ is called pointed when It is clear to see that if the ordering cone $\mathcal{C}$ is pointed, then the partial ordering “⪯” induced by $\mathcal{C}$ is antisymmetric.

Equivalently, we see that $\langle {\mathbf{A}}^{*}\rangle$ is a strongly minimal element when $\langle \mathbf{A}\rangle \in \mathcal{S}$ implies $\langle {\mathbf{A}}^{*}\rangle ⪯\langle \mathbf{A}\rangle$, which also means by referring to (8). Therefore, we can also propose the concept of strongly cone-extreme elements based on the ordering cone $\mathcal{C}$ as follows.

Next, we introduce the concept of weakly extreme elements. Let $\mathcal{S}$ be a nonempty subset of the vector space ${\mathcal{I}}_{\eta }^n$. The following set is called the algebraic interior of $\mathcal{S}$.

Let $\mathcal{C}$ be an ordering cone. Recall that $\mathcal{C}$ can induce a partial ordering defined by Using the algebraic interior $\mathrm{aint}\left(\mathcal{S}\right)$ in (10), we define which can be used to define the concept of weakly extreme elements as follows.

Suppose that $\langle {\mathbf{A}}^{*}\rangle \in \mathcal{S}$ is a weakly minimal element of $\mathcal{S}$. It means that there does not exist $\langle \mathbf{A}\rangle \in \mathcal{S}$ satisfying $\langle \mathbf{A}\rangle \prec \langle {\mathbf{A}}^{*}\rangle$, which says that there does not exist $\langle \mathbf{A}\rangle \in \mathcal{S}$ satisfying by referring to (11). Equivalently, we have

Therefore, without considering the partial ordering “⪯”, using the ordering cone, we propose the following concepts.

In this paper, we are going to study the (strongly, weakly) cone-minimal and (strongly, weakly) cone-maximal solutions of the interval-valued multiobjective optimization problems.

We consider an interval-valued function $F:X\to \mathcal{I}$ defined on a vector space X. The range of F is given by

The difference between the interval-valued functions and the functions with interval-valued coefficients can be realized from the following example.

Let us recall that, given any $A,B\in \mathcal{I}$, where $\eta :\mathcal{I}\to \mathbb{R}$ is a scalar function. Let F be an interval-valued function defined on a vector space X. Then, given any $x,y\in X$, we see that

In this case, given any fixed $z\in X$, we can define a subset $\mathcal{I}\left(z\right)$ of $\mathcal{I}$ by

This says that the range $F\left(X\right)$ can be partitioned into disjoint subsets $\mathcal{I}\left(z\right)$ such that each bounded closed interval in $\mathcal{I}\left(z\right)$ has the same scalar via the scalar function $\eta$. In other words, we have

It is clear to see that

Under the above settings, it is reasonable to define a function $\langle F\rangle :X\to {\mathcal{I}}_{\eta }$ by where $F\left(x\right)\in \mathcal{I}\left(z\right)$ for some $z\in X$ by referring to (12). If $F\left(x^{\prime }\right)\in \langle F\left(x\right)\rangle$, then

Suppose that $x=y$. Then, we have $F\left(x\right)=F\left(y\right)$, which also says

Therefore, the function $\langle F\rangle$ corresponding to F is well-defined.

Now, we consider an interval-valued vector function $\mathbf{F}:X\to {\mathcal{I}}^n$ given by where each $F_i:X\to \mathcal{I}$ is an interval-valued function defined on X for $i=1,\cdots ,n$. Therefore, each $F_i$ has a corresponding function $\langle F_i\rangle :X\to {\mathcal{I}}_{\eta }$ given by $\langle F_i\rangle \left(x\right)=\langle F_i\left(x\right)\rangle$ for $i=1,\cdots ,n$. In this case, the interval-valued vector function $\mathbf{F}$ can have a corresponding function $\langle \mathbf{F}\rangle :X\to {\mathcal{I}}_{\eta }^n$ given by

Given a partial ordering “⪯” on $\mathcal{I}$, we consider the following constrained interval-valued multiobjective programming problem where $G_i$ are interval-valued functions defined on X for $i=1,\cdots ,m$. We need to interpret the meaning of $G_i\left(x\right)⪯[0,0]$. Recall that the equivalent class $\langle 0\rangle$ is given by

According to (13), let $\langle G_i\rangle :X\to {\mathcal{I}}_{\eta }$ be a function corresponding to $G_i$ given by $\langle G_i\rangle \left(x\right)=\langle G_i\left(x\right)\rangle$ for $i=1,\cdots ,m$. Then, each constraint $G_i\left(x\right)⪯[0,0]$ is interpreted as $\langle G_i\rangle \left(x\right)⪯\langle 0\rangle$. In this case, the constrained interval-valued multiobjective programming problem can be interpreted as follows

It is clear to see that is a convex cone in the vector space ${\mathcal{I}}_{\eta }$. We see that $\langle A\rangle \in -{\mathcal{C}}_0$ means $-\langle A\rangle \in {\mathcal{C}}_0$, i.e., $\langle 0\rangle ⪯-\langle A\rangle$. Using the compatibility of vector addition for the partial ordering, we can add $\langle A\rangle$ on both sides to obtain which says that the constraint $\langle G_i\rangle \left(x\right)⪯\langle 0\rangle$ means $\langle G_i\rangle \left(x\right)\in -{\mathcal{C}}_0$ for $i=1,\cdots ,n$. In this case, the constrained interval-valued multiobjective programming problem can now be interpreted as follows: where ${\mathcal{C}}_0$ given in (15) is a special kind of ordering cone in ${\mathcal{I}}_{\eta }$.

In the sequel, we shall consider a general ordering cone $\mathcal{C}$ to study the following constrained interval-valued multiobjective programming problem:

By referring to (14), the interval-valued vector function $\mathbf{F}$ can have a corresponding function $\langle \mathbf{F}\rangle$. Therefore, we consider the following constrained interval-valued multiobjective programming problem

In order to introduce the solution concepts of problem (IMOP), we also need to consider another ordering cone in the vector space ${\mathcal{I}}_{\eta }^n$. Let $\mathcal{C}$ be an ordering cone in ${\mathcal{I}}_{\eta }^n$. This ordering cone $\mathcal{C}$ is used to rank the multiobjective function values $\langle \mathbf{F}\rangle \left(x\right)$. In this case, it is more convenient to say that the constrained interval-valued multiobjective programming problem (IMOP) is considered under the ordering cones $(\mathcal{C},\mathcal{C})$.

Let be the feasible set of problem (IMOP), and let be the set of all objective values of problem (IMOP). Then, we propose the following solution concepts.

We denote by $X^{CO}$, $X^P$, and $X^{WP}$ the set of all complete optimal solutions, Pareto optimal solutions, and weak Pareto optimal solutions of problem (IMOP), respectively. Then, we have the following inclusions.