THREE ROBUST EFFICIENCY FOR UNCERTAIN MULTIOBJECTIVE OPTIMIZATION PROBLEMS VIA SET ORDER RELATIONS

. In this paper, we propose three concepts of robust eﬃciency for uncertain multiobjective optimization problems by replacing set order rela- tions with the minmax less order relation, the minmax certainly less order relation and the minmax certainly nondominated order relation, respectively. We make interpretations for these concepts and analyze the relations between new concepts and the existent concepts of eﬃciency. Some examples are given to illustrate main concepts and results.


1.
Introduction. Optimization problems in most real world are affected by uncertain data. A decision maker needs to address this important subject in order to obtain some optimal solutions that remain feasible for an uncertain optimization problem. Sometimes, it is very important to estimate the effects of the uncertainties and so it is necessary to evaluate how sensitive an optimal solution is to perturbations of the input data. Dealing with uncertain optimization problems yields two basic approaches that are discussed in the literatures. In stochastic optimization, the uncertain parameter is assumed to obey a probability distribution and the objective is to find a feasible solution that optimizes the expected value of some objective or cost function. For an introduction to stochastic optimization, we refer to Birge and Louveaux [3]. The other way is described by robust optimization which is an active field of research. In the concept of robustness it is not assumed that all data are known, but one allows different scenarios for the input various data and looks for solutions that works well in every uncertain scenario. Robust optimization has come to encompass several approaches to protect the decision maker who must determine what it means for him or her to have a robust solution against parameter ambiguity and stochastic uncertainty.
One of the first researchers to study robust optimization problems was Soyster [23]. Minmax robustness called strict robustness was first mentioned in [23] and In order to minimize a vector valued function, we need to define the meaning of minimum on R k . We use the relations , ≤, < to compare vectors (see [5]). Let y 1 , y 2 ∈ R k , we define y 1 y 2 if y 1 is smaller or equal to y 2 in every component, y 1 ≤ y 2 if y 1 is smaller or equal to y 2 in every component and smaller in at least one component, and y 1 < y 2 if y 1 is smaller than y 2 in every component. Notice that this implies the equivalence of the relations ≤ and < in R. Additionally, we define the sets R k , R k ≥ , R k > as follows: Given a set of scenarios U ⊆ R m called uncertainty set, an uncertain optimization problem P (U ) is given as the family P (ξ)(ξ ∈ U ) of multiobjective optimization problems. Let f : R n × U → R k and X ⊆ R n , then P (ξ) is defined as the following min x∈X f (x, ξ).
We use the notation for the image of the uncertainty set U and all x ∈ X under f . Obviously for |U | = 1, P (U ) reduces to a (deterministic) multiobjective optimization problem. We will use this special case to justify and compare our concepts and results.
We introduce the concept of robust efficiency (also called the upper set less ordered efficiency in [10]) for uncertain multiobjective optimization problems in [4, Definition 3.1]: Definition 2.1. Given an uncertain multiobjective optimization problem P (U ), we call a feasible solutionx ∈ X (i) robust strictly efficient, if there is no Or alternatively, a solutionx ∈ X is called upper set less ordered (strictly/·/weakly) efficient, if there is no Let A be a nonempty subset of a partially ordered linear space Y := R k with a pointed cone R k . We introduce the minimal element and the maximal element of the set A by using the definition given in [12,Definition 4.1], then the sets of minimal elements and maximal elements of A can be characterized as We will use the following notations: and for all x ∈ X, the latter is defined in [10,Lemma 31]. Now we recall the set less order relation, the set less ordered efficiency, the certainly less order relation and the certainly less ordered efficiency used in this paper: Definition 2.2. [14] Let A, B ∈ P(Y ) be arbitrarily chosen sets. Then A is said to dominate B with respect to the set less order relation (we denote that by A s B) and with respect to Definition 2.3.
[10] Given an uncertain multiobjective optimization problem P (U ), a solutionx ∈ X is called set less ordered (strictly/·/weakly) efficient for , which is equivalent to Definition 2.4. [14] Let A, B ∈ P(Y ) be arbitrarily chosen sets. Then A is said to dominate B with respect to the certainly less order relation (we denote that by A c B) and with respect to Definition 2.5. [10] Given an uncertain multiobjective optimization problem P (U ), a solutionx ∈ X to P (U ) is called certainly less ordered (strictly/·/weakly) efficient, if there is no x 0 ∈ X\{x} such that Remark 1. From Lemma 31 and Remark 32 of [10] we see that, for allx,x ∈ X, In addition, the lower set less ordered efficiency and the alternative set less ordered efficiency were defined in [10, Definitions 9 and 26], which are given in the following: Definition 2.6. [10] Given an uncertain multiobjective optimization problem P (U ), a solutionx ∈ X is called lower set less ordered (strictly/·/weakly) efficient, if there is no Definition 2.7.
[10] Given an uncertain multiobjective optimization problem P (U ), a solutionx ∈ X of P (U ) is called alternative set less ordered (strictly/·/weakly) efficient, if there is no 3. New definitions of robust efficiency. Ide and Köbis [10] derived new concepts of efficiency for uncertain multiobjective optimization problems by replacing the set ordering with others, such as lower set less ordered efficiency, set less ordered efficiency, alternative set less ordered efficiency and certainly less ordered efficiency. In addition, they gave a detail interpretation and discussed the relations between these concepts. Besides, Köbis [18] introduced the minmax less ordered robustness and the minmax certainly less ordered robustness which made the robustness concepts abundant. In [4] the concept of robust efficiency is closely connected to the upper set less ordering introduced by Kuroiwa [20]. Based on the study of [10,4,18] and new order relations defined by Jahn and Ha in [14], we propose three definitions of efficiency for uncertain multiobjective optimization. Compared with the existing concepts, we replace the sets f U (x) by their minimal and maximal elements, which seems to be more clear to express the decision maker's preferences.
3.1. Minmax less ordered efficiency. The first proposed concept of efficiency for uncertain multiobjective optimization problems is called minmax less ordered efficiency which is useful and significant for decision making. Now we introduce the minmax less order relation defined by Jahn and Ha [14, Definition 3.5].
Definition 3.1. Let A, B ∈ M be arbitrarily chosen sets. We say that A dominates B with respect to the minmax less order relation (we denote this by Given an uncertain multiobjective optimization problem P (U ), for all Considering the uncertain multiobjective optimization problem and combining with Definition 2.2, we have the following set relation: for all x,x ∈ X, . Now we define the minmax less ordered efficient solutions for an uncertain multiobjective optimization problem.
Definition 3.2. Given an uncertain multiobjective optimization problem P (U ), a solutionx to P (U ) is called minmax less ordered (strictly/·/weakly) efficient, if there is no The concept of minmax less ordered efficiency is appealing for a decision maker because its definition contains comparisons of minimal as well as maximal elements of sets. In that manner it reflects optimism about the future as well as the risk averse of the approaches containing maximal elements. Contrary to upper (lower) set less ordered efficiency, the decision maker is now able to hedge against the minimal (maximal) solutions of sets f U (x) instead of the whole lower (upper) bound. This enables a user to specify his/her wishes during the decision process even more. This concept is less restrictive than set less ordered efficiency.
Remark 2. This concept, compared with [18,Definition 20], uses a specific cone R k to define the minimal and maximal elements of f U (x). Besides, the definition of minmax less ordered (·/weakly) efficient solution is also different. By this means, it may be consistent with the existing concepts [10,11] and convenient to investigate their relations.
An example is given to interpret the minmax less ordered efficiency. Example 1. Consider Figure 1. An uncertain multiobjective optimization problem P (U ) with feasible set X = {x 1 , . . . , x 5 } and the five sets f U (x 1 ), . . . , f U (x 5 ) are described as polygons. In Figure 2 and Figure 3, the left pictures show the sets of their Min and Max. By adding −R 2 and R 2 to each of these sets, in the middle and right pictures we can see that , respectively), thus x 1 and x 3 are the minmax less ordered strictly efficient solutions.
We check if the minmax less ordered efficiency is consistent in the case P (U ) with |U | = 1. It is obvious that this result is a special case of [18, Lemma 12].

3.2.
Minmax certainly less ordered efficiency. The next introduced concept is called minmax certainly less ordered efficiency which is helpful for us to have a better understanding of a decision maker's preferences. Before the concept proposed, we firstly introduce the minmax certainly less order relation defined by Jahn and Ha [14,Definition 3.6]. Due to Definitions 2.4 and 3.3, we are able to define the minmax certainly less ordered efficient solutions.
Definition 3.4. Given an uncertain multiobjective optimization problem P (U ), a solutionx to P (U ) is called minmax certainly less ordered (strictly/·/weakly) efficient with respect to The concept of minmax certainly less ordered efficiency based on the minmax certainly less order relation is less restrictive than the minmax less ordered efficiency and demands some solutions which strictly dominate others in both the minimal and maximal elements of f U (x) (the best and worst scenarios). This concept would reflect a decision maker's preferences precisely if he/she is optimistic or risk-averse about the future.
Remark 3. This concept, especially the (·/weakly) efficient solution of Definition 3.4, is different from [18,Definition 25] because of the exact definitions of minimal and maximal elements of f U (x). Certainly, we could be able to discuss the relations between this concept and those ones in [10,11]. Example 2. See Example 1, since x 0 ∈ X\{x 1 } (X\{x 2 }, X\{x 3 }, respectively) such that then x 1 , x 2 and x 3 are minmax certainly less ordered strictly efficient.
Again, for P (U ) with |U | = 1 reduced to a deterministic multiobjective optimization problem, the minmax certainly less ordered efficiency is a special case of [18,Lemma 19].
3.3. Minmax certainly nondominated ordered efficiency. The last introduced concept which is called minmax certainly nondominated ordered efficiency intends to filter out some bad solutions relatively. In order to propose this concept, we give the minmax certainly nondominated order relation defined by Jahn and Ha [14,Definition 3.8]. Combined Definition 2.2 with Definition 3.5, we define the minmax certainly nondominated ordered efficient solutions. Definition 3.6. Given an uncertain multiobjective optimization problem P (U ), a solutionx is called minmax certainly nondominated ordered (strictly/·/weakly) efficient with respect to . The purpose of this concept is to sort out some solutions in the best scenarios which are dominated by others in the worst scenarios. If a design maker is risk-averse, those selected solutions are filtered out primarily. By this concept, it intuitively reflects some degree of a design maker's risk-aversion. Additionally, this concept is less restrictive than the minmax less ordered efficiency and seems to be a completely new definition.
then x 1 , x 2 and x 3 are minmax certainly nondominated ordered strictly efficient.
Furthermore, we check the concept of minmax certainly nondominated ordered efficiency for consistency in the case that P (U ) is a deterministic multiobjective optimization problem: Proposition 1. Given an uncertain multiobjective optimization problem P (U ) with |U | = 1, then a solutionx ∈ X is the minmax certainly nondominated ordered (strictly/·/weakly) efficient iff it is (strictly/·/weakly) efficient.

4.
Relationships between robust efficient concepts. In Section 3, we propose three definitions of efficiency for uncertain multiobjective optimization problems and give interpretations for them. At present, some of the existing concepts, i.e., [lower/upper/alternative/·] set less ordered efficiency and certainly less ordered efficiency were introduced in [4,10]. Before discussing the relations between new concepts and the existing concepts, we introduce the quasi domination property which relates a set and the sets of its minimal and maximal elements given in [14, Definition 3.7].

Definition 4.1. A set A ∈ M is said to have the quasi domination property iff
We assume that f U (x) has the quasi domination property for all x ∈ X. From Definition 4.1 we know that ∀x ∈ X, which will be used in the proofs of Theorems 4.2 and 4.5. Firstly, we explore the relation between the set less ordered efficiency and the minmax less ordered efficiency. From [18,Theorem 33] we know that a set less ordered strictly efficient solution must be minmax less ordered strictly efficient. The following theorem implies that if a solution is set less ordered (weakly) efficient, then it is minmax less ordered (weakly) efficient.
Theorem 4.2. Given an uncertain multiobjective optimization problem P (U ), if a solutionx ∈ X is set less ordered (weakly) efficient, then it is minmax less ordered (weakly) efficient.
Proof. It is analogous to the proof of Theorem 33 in [18] because of the equation Unfortunately, the converse of Theorem 4.2 is, in general, not true. The following example illustrates this result.
Example 4. Consider Figure 4. An uncertain multiobjective optimization problem P (U ) with feasible set X = {x 1 , x 2 } and two sets f U (x 1 ), f U (x 2 ) are depicted as rectangle and circle respectively. We can easily obtain f U (x 1 ) s f U (x 2 ) with respect to R 2 ≥ , thus x 1 is a set less ordered efficient solution. Since Maxf U (x 1 ) s Maxf U (x 2 ), then x 1 , x 2 are minmax less ordered efficient. However, x 2 is not set less ordered efficient. Based on Theorems 39 and 40 in [18], a minmax less ordered strictly efficient solution is minmax certainly less ordered strictly efficient and a minmax certainly less ordered strictly efficient solution implies a certainly less ordered strictly efficient solution. Next we study the relations between their (weakly) efficient solutions.  Given an uncertain multiobjective optimization problem P (U ), if a solutionx ∈ X is minmax certainly less ordered (weakly) efficient, then it is certainly less ordered (weakly) efficient.
Proof. Combined Remark 1 and the proof of [18,Theorem 40], the conclusion is true.
Generally, the converses of Theorems 4.3 and 4.4 are not true.
, then x 1 , x 2 , x 3 and x 4 are certainly less ordered efficient. From Examples 1 and 2 we have known that x 2 is minmax certainly less ordered efficient but not minmax less ordered efficient, x 4 is certainly less ordered efficient rather than minmax certainly less ordered efficient. Therefore, Theorems 4.3 and 4.4 just provide sufficient conditions. From [14, Proposition 3.10], we have known the relations between the minmax less order relation, the minmax certainly nondominated order relation and the certainly less order relation. We investigate the relationships among their concepts of efficiency as follows.
Proof. Ifx is not minmax certainly nondominated ordered (strictly/·/weakly) efficient, there exists an x 0 ∈ X\{x} such that combined with (1) and (2), we have . Therefore,x is not minmax less ordered (strictly/·/weakly) efficient, we arrive at a contradiction to the assumption. Theorem 4.6. Given an uncertain multiobjective optimization problem P (U ), if a solutionx ∈ X is minmax certainly nondominated ordered (strictly/·/weakly) efficient, then it is certainly less ordered (strictly/·/weakly) efficient.
Proof. Suppose thatx is not certainly less ordered (strictly/·/weakly) efficient. Then there exists an x 0 ∈ X\{x} such that in contradiction to the assumption ofx being minmax certainly nondominated ordered (strictly/·/weakly) efficient for P (U ).
Normally the converses of Theorems 4.5 and 4.6 are also not true.
Example 6. See Example 1, combined with Examples 3 and 5, we know that x 2 is minmax certainly nondominated ordered strictly efficient but not minmax less ordered strictly efficient and x 4 is certainly less ordered strictly efficient instead of a minmax certainly nondominated ordered strictly efficient solution.
In the end, we discuss the relation between two concepts of the minmax certainly less ordered efficiency and the minmax certainly nondominated ordered efficiency. If a solution is minmax certainly less ordered (strictly/·/weakly) efficient, it is not necessarily minmax certainly nondominated ordered (strictly/·/weakly) efficient. Moreover, a minmax certainly nondominated ordered (strictly/·/weakly) solution may not be minmax certainly less ordered (strictly/·/weakly) efficient. Let us see the following two examples.
Example 7. Consider Figure 5. An uncertain multiobjective optimization problem P (U ) with feasible set X = {x 1 , x 2 } and two sets f U (x 1 ), f U (x 2 ) are depicted as circle and triangle respectively. We can easily obtain f U (x 1 ) mn f U (x 2 ) with respect to R 2 , thus x 1 is a minmax certainly nondominated ordered strictly efficient , thus x 1 , x 2 are minmax certainly less ordered strictly efficient. But x 2 is not minmax certainly nondominated ordered strictly efficient.   [10,18], we supplement relationships between new concepts and the existing concepts in Figure 7, where "IK" stands for Ref. [10]. 5. An Application for Tourist's destination selection. With the rapid development of economy and the improvement of living level of people, more and more people go traveling on holidays. It is very significant for travellers to select suitable tourist destinations. In general, they value the most aspects that are the entertainment of tourist spots and the amount of tourist crowds as two objective functions. They want to maximize the former and minimize the latter. It is very difficult to make choices for them. For one thing they are not sure what the weather will be and how many tourists will visit the specific sight, for another with different weather conditions, the entertainment may be different and visiting a specific tourist spot may have less fun.
They evaluated the different tourist spots and edited the data in the following way: Assume there are four weather scenarios, each of them resulting in a different score on entertainment factor and tourist crowds for each sight. The score is estimated in grades from 1 to 20, 1 being perfect and 20 being very bad. We denote entertainment factor and tourist crowds as EF and TC respectively. Table 1 provides the results and S i , i ∈ I := {1, . . . , 10} denote the tourist destinations respectively (A part of the data of Table 1 is from Table 1 in Ref. [10], the remaining data is estimated based on the fact.). Since this problem is multiobjective, they have to make trade-offs between the two objective functions. Moreover, since the problem is uncertain, they need to define what would be a suitable solution and make a decision through considering all four weather scenarios. Now, we will apply these concepts of robustness to analyse this problem. Figure  8 shows the objective values of Table 1. From Figure 8, we can obtain that S i , i = 1, 4, 5, 6 are minmiax less ordered strictly efficient and S i , i ∈ I\{2, 3, 8, 9} are minmiax certainly less ordered strictly efficient. See Figure 9, it is obvious that S i , i ∈ I\{2, 9} are minmiax certainly nondominated ordered strictly efficient and S i , i ∈ I\{2} are certainly less ordered strictly efficient.  Table 1 We can obtain different solutions by using different robustness concepts, which not only reflects visitors' preferences but also is more convenient and helpful for them to select appropriate tourist destinations. Furthermore, if they prefer to the entertainment of sights than the amount of tourist crowds, it is easier to make a choice, and vice versa. In addition, if EF and TC, weather scenarios and tourist destinations are substituted by cost and transportation speed, transportation means and warehouse locations respectively, these concepts can be applied to some special  Figure 9. Comparisons of solutions facility location problems. If we replace EF and TC, weather scenarios and tourist destinations by profit and risk, capital grades and investment proposals respectively, we can also use these concepts to portfolio problems. 6. Conclusions. In this paper, we introduced three concepts of robust efficiency which differ from the existent concepts in [10,18] for uncertain multiobjective optimization problems. We made explanations for these concepts and discussed the relationships between new concepts and the existing concepts of efficiency. Some examples were also provided to illustrate main concepts and results. These concepts serve as a supplement to the existent concepts in [10] and make the concepts of efficiency for uncertain multiobjective optimization problems complete. In order to estimate the usefulness of robust efficient solutions, more applications of the presented concepts for real world problems are worth studying. Specifically, the robust solution and its properties for parametric generalized vector equilibrium problems [26] should be focused. Additionally, when we finish this work, we find that Ide and Schöbel [11] have presented and compared ten different concepts of robustness (including all known ones in Ide and Köbis [10]) for uncertain multi-objective optimization very recently. Combining with the concepts introduced herein, more relationships between these various concepts may be revealed in future work.