In this section, we develop a novel approach on the basis of preferencebased index to convert and cope with the interval bilevel linear programming problem (1).
4.1 Conversion of the interval constraint region into its crisp one
To deal with problem (1), we first shall discuss crisp equivalent transformation for the interval constraint region in problem (1). In order to do so, we first need to convert each interval inequality constraint into its crisp form. From the point of decision maker’s satisfaction, several typical methods introduced by Sengupta et al. [33], Guo and Wu [34], and Allahdadi and Nehi [35] mainly focus on transforming an interval inequality constraint into a type of satisfactory crisp constraint. However, it has been pointed out by Chen et al. [27] that these types of conversion could lose the uncertainty of intervals in part during the transformation process. In a different approach, recently Chen et al. [27] discussed a new equivalent transformation for interval inequality constraint in terms of normal variation of interval number and chanceconstrained programming approach. The main advantage of this kind of transformation is to maintain as much uncertainty as possible. In this way, we first transform any interval inequality constraint of problem (1) into a stochastic inequality form by normal variation of intervals.
For any interval \(a^{I}=[a^{},a^{+}]\), in terms of the 3σ law, its corresponding normally distributed random variable, denoted by \(\bar {a}\sim N(\mu,\sigma^{2})\), can be determined as follows:
$$\mu=\frac{a^{}+a^{+}}{2},\qquad \sigma=\frac{a^{+}a^{}}{6}. $$
Clearly, we have \([a^{},a^{+}]=[\mu3\sigma,\mu+3\sigma]\). Notice that the probability that any interval number ā falls in \([a^{},a^{+}]\) is 99.73% on the basis of the 3σ law. Thus it is reasonable to utilize a normally distributed random variable to represent an interval.
By replacing interval coefficients with their corresponding normally distributed random variables, the ith interval inequality constraint of problem (1) can be converted as:
$$\bar{a}_{i1}x_{1}+\bar{a}_{i2}x_{2}\geq \bar{b}_{i},\quad i=1,2,\ldots,m, $$
where \(\bar{a}_{ij}=(\bar{a}_{ij1},\bar{a}_{ij2},\ldots,\bar {a}_{ijn_{j}})\), \(i=1,2,\ldots,m\), \(j=1,2\), are corresponding normally distributed random vectors of interval vectors \([a_{ij}^{},a_{ij}^{+}]\), and \(\bar{b}_{i}\), \(i=1,2,\ldots,m\) are corresponding normally distributed random variables of interval numbers \([b_{i}^{},b_{i}^{+}]\). For simplicity, we assume that \(\bar {a}_{ij1},\bar{a}_{ij2},\ldots,\bar{a}_{ijn_{j}}\) and \(\bar{b}_{i}\) are independent from each other.
As is well known, chanceconstrained programming approach [36] is the most applied one to handle the stochastic constraints. By this means, stochastic constraints hold at least some satisficing probability level specified by decision makers. With the aid of chanceconstrained programming, the above ith stochastic inequality constraint can be reformulated as follows:
$$\operatorname{Pr}\{\bar{a}_{i1}x_{1}+\bar{a}_{i2}x_{2} \geq\bar{b}_{i}\}\geq \beta_{i},\quad i=1,2,\ldots,m, $$
where Pr denotes the probability of the event, and \(\beta_{i}\in(0,1)\) is the probability level of the ith constraint, \(i=1,2,\ldots,m\).
In terms of Theorem 5.1 in [27], the crisp equivalent form of the above ith chance inequality constraint can be obtained.
Now we have converted the interval constraint region of problem (1) into its crisp structure. Then problem (1) can be transformed into the following problem:
$$ \textstyle\begin{cases} {\min_{x_{1}}} & [c^{}_{11},c^{+}_{11}]x_{1}+[c^{}_{12},c^{+}_{12}]x_{2}\\ & \text{where } x_{2} \text{ solves}\\ {\min_{x_{2}}} & [c^{}_{21},c^{+}_{21}]x_{1}+[c^{}_{22},c^{+}_{22}]x_{2}\\ \mathrm{s.t.} & \sum_{s=1}^{n_{1}}E(\bar{a}_{i1s})x_{1s}\\ &\qquad {}+\sum_{t=1}^{n_{2}}E(\bar{a}_{i2t})x_{2t}\\ &\qquad {}+\Phi^{1}(\beta_{i})\sqrt{\sum_{s=1}^{n_{1}}V(\bar{a}_{i1s})x^{2}_{1s}+\sum_{t=1}^{n_{2}}V(\bar {a}_{i2t})x^{2}_{2t}+V(\bar{b}_{i})}\\ &\quad \geq E(\bar{b}_{i}),\quad i=1,2,\ldots,m,\\ & x_{1}=(x_{11},x_{12},\ldots,x_{1n_{1}})^{\mathrm{ T}}\geq 0,x_{2}=(x_{21},x_{22},\ldots,x_{2n_{2}})^{\mathrm{ T}}\geq0, \end{cases} $$
(2)
where \(E(\cdot)\) and \(V(\cdot)\) denote the expectation and variance of a random variable, and \(\Phi^{1}\) is the inverse function of standardized normal distribution.
For convenience, we denote the constraint region of problem (2) by D.
Equivalently, problem (2) can be rewritten as
$$ \textstyle\begin{cases} {\min_{x_{1}}} & F^{I}(x_{1},x_{2})=[c^{}_{11},c^{+}_{11}]x_{1}+[c^{}_{12},c^{+}_{12}]x_{2}\\ & \text{where } x_{2} \text{ solves}\\ {\min_{x_{2}}} & f^{I}(x_{1},x_{2})=[c^{}_{21},c^{+}_{21}]x_{1}+[c^{}_{22},c^{+}_{22}]x_{2}\\ \mathrm{s.t.} & (x_{1},x_{2})\in D, \end{cases} $$
(3)
where \(F^{I}=[F^{},F^{+}]\) and \(f^{I}=[f^{},f^{+}]\) denote interval objective functions of the upper and lower level programming problems, respectively.
Clearly, the above problem is a bilevel programming problem with interval coefficients in both objective functions only.
4.2 Preferencebased deterministic bilevel programming problem
In order to reflect different preferences of different decision makers, in this section we first introduce the concept of the preference level that the interval objective function is preferred to a target interval in light of preferencebased index in [28]. Then we build a preferencebased deterministic bilevel programming problem for problem (3) based on the preference level and the order relation \(\preceq_{mw}\).
For a given \(x_{1}\), the lower level programming problem of problem (3) is an essentially single level optimization problem for objective function involving interval coefficients. For this type of problem, the midpoints of the intervals are often used to cope with the related interval objective functions, but it may cause the loss of helpful information to some extent in terms of such a treatment. To better reflect uncertain information, the order relation \(\leq_{mw}\) which considers the midpoint and halfwidth of intervals at the same time is employed to compare different interval objective function values of the lower level problem for different decision variables. For given \(x_{1}\), denote the feasible region of the lower level problem by \(D(x_{1})\). Based on the linear combination method, then the lower level problem can be converted into the following problem:
$$ \textstyle\begin{cases} {\min_{x_{2}}} & \theta m(f^{I}(x_{1},x_{2}))+(1\theta )w(f^{I}(x_{1},x_{2}))\\ & x_{2}\in D(x_{1}), \end{cases} $$
(4)
where \(\theta (0\leq\theta\leq1)\) denotes a weighting factor, \(m(f^{I}(x_{1},x_{2}))=\frac {f^{}(x_{1},x_{2})+f^{+}(x_{1},x_{2})}{2}\) and \(w(f^{I}(x_{1}, x_{2}))=\frac{f^{+}(x_{1},x_{2})f^{}(x_{1},x_{2})}{2}\) represent the midpoint and halfwidth of the lower level objective function \(f^{I}(x_{1},x_{2})\), respectively.
Denote the set of optimal solutions of problem (4) by \(M^{I}(x_{1})\) for a given \(x_{1}\). Then the feasible region of problem (3) can be defined as:
$$IR^{I}=\bigl\{ (x_{1},x_{2})(x_{1},x_{2}) \in D, x_{2}\in M^{I}(x_{1})\bigr\} . $$
Thus problem (3) can be formulated as:
$$ \textstyle\begin{cases} {\min_{x_{1},x_{2}}} & F^{I}(x_{1},x_{2})\\ \mathrm{s.t.} & (x_{1},x_{2})\in IR^{I}. \end{cases} $$
(5)
It is well known that the order relation between intervals is often applied to tackle interval objective function. So far there have been all sorts of definitions of the interval order relations based on various mathematical approaches [37–39] with the exception of the order relation \(\preceq_{mw}\) mentioned above. Among them, Sengupta and Pal [38] proposed the acceptability index for ranking any two interval numbers based on decision maker’s satisfaction. It has been pointed out by Ruan et al. [28] that this index cannot be applied for reflecting different preferences from different decision makers. For this purpose, we apply preferencebased index for ranking interval numbers introduced by Ruan et al. [28] to treat the interval objective function in the upper level of problem (5).
In order to find a satisfying solution for the decision maker at the upper level, the interval objective function often needs to satisfy a desired target interval determined in advance by the decision maker as far as possible. Let \(C^{I}=[C^{},C^{+}]\) be a predetermined target interval corresponding to the upper level objective function. Based on preferencebased index for ranking interval numbers, we define the concept of the preference level as follows.
Definition 4
For any \((x_{1},x_{2})\geq0\), \(\delta (F^{I}(x_{1},x_{2})\prec C^{I})\) is said to be a preference level that the interval objective function \(F^{I}(x_{1},x_{2})\) is inferior to a target interval \(C^{I}\). If \(\delta(F^{I}(x_{1},x_{2})\prec C^{I})>0\), \(F^{I}(x_{1},x_{2})\) is preferred to \(C^{I}\) for a minimization problem.
Clearly, it needs to maximize the preference level of the interval objective function to be inferior to its target interval for a minimization problem. In this sense, problem (5) can be transformed into the following preferencebased deterministic bilevel programming problem:
$$ \textstyle\begin{cases} {\max_{x_{1},x_{2}}} & \delta(F^{I}(x_{1},x_{2})\prec C^{I})\\ \mathrm{s.t.} & (x_{1},x_{2})\in IR^{I}. \end{cases} $$
(6)
Now we give the concept of a preference δoptimal solution for the interval bilevel linear programming problem (1).
Definition 5
A solution \((x^{*}_{1},x^{*}_{2})\in IR^{I}\) is said to be a preference δoptimal solution of problem (1) if there does not exist another feasible solution \((x_{1},x_{2})\in IR^{I}\) such that \(\delta(F^{I}(x_{1},x_{2})\prec C^{I})\geq\delta (F^{I}(x^{*}_{1},x^{*}_{2})\prec C^{I})\).
Using Definition 3, we have
$$\delta\bigl(F^{I}(x_{1},x_{2})\prec C^{I}\bigr)=\frac {o(C^{I})o(F^{I}(x_{1},x_{2}))}{w(C^{I})+w(F^{I}(x_{1},x_{2}))+1}. $$
Thus problem (6) can be rewritten as
$$ \textstyle\begin{cases} {\max_{x_{1}}} & \frac {o(C^{I})o(F^{I}(x_{1},x_{2}))}{w(C^{I})+w(F^{I}(x_{1},x_{2}))+1}\\ & \text{where } x_{2} \text{ solves}\\ {\min_{x_{2}}} & \theta m(f^{I}(x_{1},x_{2}))+(1\theta )w(f^{I}(x_{1},x_{2}))\\ & \sum_{s=1}^{n_{1}}E(\bar{a}_{i1s})x_{1s}\\ &\qquad{}+\sum_{t=1}^{n_{2}}E(\bar {a}_{i2t})x_{2t}\\ &\qquad{}+\Phi^{1}(\beta_{i})\sqrt{\sum_{s=1}^{n_{1}}V(\bar {a}_{i1s})x^{2}_{1s}+\sum_{t=1}^{n_{2}}V(\bar {a}_{i2t})x^{2}_{2t}+V(\bar{b}_{i})}\\ & \quad\geq E(\bar{b}_{i}),\quad i=1,2,\ldots,m,\\ & x_{1}=(x_{11},x_{12},\ldots,x_{1n_{1}})^{\mathrm{ T}}\geq 0,x_{2}=(x_{21},x_{22},\ldots,x_{2n_{2}})^{\mathrm{ T}}\geq0. \end{cases} $$
(7)
On the other hand, if the original problem is a type of maximization stated as follows:
$$ \textstyle\begin{cases} {\max_{x_{1}}} & [c^{}_{11},c^{+}_{11}]x_{1}+[c^{}_{12},c^{+}_{12}]x_{2}\\ & \text{where } x_{2} \text{ solves}\\ {\max_{x_{2}}} & [c^{}_{21},c^{+}_{21}]x_{1}+[c^{}_{22},c^{+}_{22}]x_{2}\\ \mathrm{s.t.} & [a^{}_{i1},a^{+}_{i1}]x_{1}+[a^{}_{i2},a^{+}_{i2}]x_{2}\geq [b^{}_{i},b^{+}_{i}],\quad i=1,2,\ldots,m,\\ & x_{1}\geq0,x_{2}\geq0. \end{cases} $$
(8)
For problem (8), the decision maker hopes to maximize the preference level of the interval objective function to be superior to its target interval. By using the similar idea for the above minimization, another preferencebased deterministic bilevel programming problem of problem (8) can be expressed as:
$$ \textstyle\begin{cases} {\max_{x_{1}}} & \delta(F^{I}(x_{1},x_{2})\succ C^{I})\\ & \text{where } x_{2} \text{ solves}\\ {\max_{x_{2}}} & \theta m(f^{I}(x_{1},x_{2}))+(1\theta )(w(f^{I}(x_{1},x_{2})))\\ & (x_{1},x_{2})\in D. \end{cases} $$
(9)
Equivalently, problem (9) can be rewritten in the form
$$ \textstyle\begin{cases} {\max_{x_{1}}} & \frac {o(F^{I}(x_{1},x_{2}))o(C^{I})}{w(F^{I}(x_{1},x_{2}))+w(C^{I})+1}\\ & \text{where } x_{2} \text{ solves}\\ {\max_{x_{2}}} & \theta m(f^{I}(x_{1},x_{2}))+(1\theta )(w(f^{I}(x_{1},x_{2})))\\ & \sum_{s=1}^{n_{1}}E(\bar{a}_{i1s})x_{1s}\\ &\qquad{}+\sum_{t=1}^{n_{2}}E(\bar {a}_{i2t})x_{2t}\\ &\qquad{}+\Phi^{1}(\beta_{i})\sqrt{\sum_{s=1}^{n_{1}}V(\bar {a}_{i1s})x^{2}_{1s}+\sum_{t=1}^{n_{2}}V(\bar {a}_{i2t})x^{2}_{2t}+V(\bar{b}_{i})}\\ &\quad \geq E(\bar{b}_{i}),\quad i=1,2,\ldots,m,\\ & x_{1}=(x_{11},x_{12},\ldots,x_{1n_{1}})^{\mathrm{ T}}\geq 0,x_{2}=(x_{21},x_{22},\ldots,x_{2n_{2}})^{\mathrm{ T}}\geq0. \end{cases} $$
(10)
Clearly, problems (7) and (10) are a class of nonlinear bilevel programming problems which are coped with by one of the metaheuristic techniques namely estimation of distribution algorithm in the next subsection.
4.3 Solution approach based on estimation of distribution algorithm
It is well known that bilevel programming problem is a complex optimization model and it is difficult to tackle. Usually, traditional solution methods involve huge computational load when solving this type of problem and they are only successful for some special bilevel cases. Estimation of distribution algorithm (EDA) [40], which is a new evolutionary metaheuristic algorithm, has attracted considerable attention as an alternative method for solving bilevel programming problem [41, 42] in recent years. For EDA, the main steps of the iterative procedure include: randomly create initial population, select some excellent individuals, build a probabilistic model based on excellent individuals chosen, generate new individuals by sampling from the constructed probabilistic model, and repeat the cycle until a stopping criterion is met. Notice that the main characteristics of this approach is to reproduce a new generation implicitly by sampling from a probability model constructed by promising candidate solutions.
In our work, estimation of distribution algorithm is applied to solve nonlinear bilevel programming problems (7) and (10). For these two problems, observing that the lower level objective functions are linear and constraint functions are quadratic, thus a number of traditional techniques can be employed to solve the lower level problem. Furthermore, estimation of distribution algorithm is used to deal with the upper level problem. Based on these ideas, we give the details of the computational method by combining estimation of distribution algorithm with some traditional method for solving problems (7) and (10) as follows:

Step 1
Ask the decision makers to specify the probability levels \(\beta_{i}, i=1,2,\ldots,m\), weighting factor θ, target interval \(C^{I}\) and optimism degree of the upper level decision maker γ. Generate the initial population \(\mathrm{Pop}(0)\) with population size N comprised by the upper level decision variable. Let \(t = 0\).

Step 2
For each given upper level individual, we solve the lower level problem by means of some traditional method.

Step 3
Evaluate the fitness value defined by the upper level objective function value for each individual.

Step 4
Select M best individuals from \(\mathrm{Pop}(t)\) to form the parent set \(Q(t)\) by the truncation selection, and update the probability model by estimating the distribution of \(Q(t)\).

Step 5
Sample N new individuals from the updated probabilistic model. Denote the set of all these individuals by \(O(t)\). Select N best offspring candidates from the set \(\mathrm{Pop}(t)\cup O(t)\) to construct the next population \(\mathrm{Pop}(t+1)\).

Step 6
If the algorithm is executed to the maximal number of generations, then stop; otherwise, let \(t=t+1\), go to Step 2.