- Research
- Open Access
Robust optimization model for uncertain multiobjective linear programs
- Lei Wang^{1}Email author and
- Min Fang^{1}
https://doi.org/10.1186/s13660-018-1612-3
© The Author(s) 2018
- Received: 6 October 2017
- Accepted: 3 January 2018
- Published: 18 January 2018
Abstract
In this paper, we consider the multiobjective linear programs where coefficients in the objective function belong to uncertainty sets. We introduce the concept of robust efficient solutions to uncertain multiobjective linear programming problems. By using two scalarization methods, the weighted sum method and the ϵ-constraint method, we obtain that the robust efficient solutions for uncertain multiobjective linear programs with ellipsoidal uncertainty sets and general norm uncertainty sets can be computed by some deterministic optimization problems.
Keywords
- multiobjective linear programs
- robust efficient solutions
- scalarization
- uncertainty sets
MSC
- 90A14
- 47H04
- 47S40
1 Introduction
The parameter values of optimization problems in real world are usually uncertain due to prediction errors, estimation errors, or lack of information at the time of decision. Therefore, it is important to solve such uncertain optimization problems for decision maker. In 1973, Soyster [1] first introduced the robust linear programs where coefficients are uncertain. The main idea is to assume that the coefficients can be any scenario in the uncertainty set and to find a solution that is feasible for all possible scenarios from the uncertainty set. The interest of robust optimization was revived in the 1990s (see, e.g., [2–10]). In 2009, Ben-Tal, El-Ghaoui, and Nemirovski [11] introduced a number of important results in robust linear optimization, robust conic optimization, and robust multistage optimization. Robust optimization has become a powerful approach to handle uncertain optimization problems.
On the other hand, the equilibrium problem provides a general mathematical model for a wide range of practical problems, such as optimization problems, Nash equilibria problems, fixed point problems, variational inequality problems, and complementarity problems, and has been investigated intensively. For more details, we refer to [12–15]. As a particular case of the vector equilibrium problem, multiobjective optimization problems arise in a large number of applications such as transportation, finance, communication, etc. Naturally, the issue of uncertain data affects single objective optimization problems in the same way as it affects these multiobjective ones. The essential problem in multiobjective optimization is to find the Pareto efficient solutions, meaning the feasible solutions such that no objective can be improved without sacrificing others; see, for example, Miettinen [16] and Ehrgott [17]. Therefore, in multiobjective optimization problems with data uncertainty, it is very important how to find robust efficient solutions that are less sensitive to small perturbations in variables.
In 2006, Deb and Gupta [18] presented two different robust multiobjective optimization procedures. The first one replaces all objective functions by their mean functions, and robust solution is defined as the efficient solution to the resulting deterministic optimization problem. The second one adds constraints to the predefined limit and optimizes the original objective functions. Recently, Kuroiwa and Lee [19] defined three kinds of robust efficient solutions, which are different from Deb and Gupta [18] for the uncertain multiobjective optimization problems. They also established necessary optimality theorems for robust efficient solutions and gave scalarization methods for robust efficient solutions of multiobjective optimization problems. Ehrgott, Ide, and Schobel [20] generalized the concept of minimax robustness introduced by Ben-Tal, El-Ghaoui, and Nemirovski [11] from single objective optimization problems to multiobjective optimization problems. They proposed robust Pareto efficiency and discussed how to find robust efficient solutions for uncertain multiobjective optimization problems. Goherna, Jeyakumar, Li, and Perez [21] introduced the definition of radius of robust feasibility and analyzed the robust weakly efficient solution of a multiobjective linear programming problems with data uncertainty both in the objective function and constraints. They also gave numerically tractable optimality conditions for highly robust weakly efficient solutions. Very recently, Bokrantz and Fredriksson [22] provided necessary and sufficient conditions for robust efficiency studied by Ehrgott, Ide, and Schobel [20] to multiobjective optimization problems with data uncertainty. They also applied these results to the field of therapy for cancer treatment.
Motivated by the works mentioned, in this paper, we consider the multiobjective linear programs where the coefficients \(a_{i}\) and \(b_{i}\) in the objective function belong to uncertain but bounded sets \(U_{i}\). We introduce the concept of robust efficient solution to uncertain multiobjective linear programs (UMLPs). Also, we show that two scalarization methods, the weighted sum method and the ϵ-constraint method, can be used to find robust efficient solutions of UMLP with ellipsoidal uncertainty sets and general norm uncertainty sets. The structure of the paper is as follows. In Section 2, we introduce the uncertain multiobjective linear programming problems and the concept of robust efficient solution to UMLP. In Section 3, we give the ellipsoidal uncertainty sets and general norm uncertainty sets. Using the weighted sum method, we show that the robust efficient solution of UMLP can be found by some deterministic optimization problems. In Section 4, we use the ϵ-constraint method to compute the robust efficient solution of UMLP under ellipsoidal uncertainty sets and general norm uncertainty sets. Finally, we conclude in Section 5.
2 Introduction to uncertain multiobjective linear programs
2.1 Deterministic multiobjective linear programs
In this paper, we use the order relation ⪰ (see Ehrgott [17]): For \(y^{1}, y^{2}\in R^{m}\), we write \(y^{1}\succeq y^{2}\) if \(y^{1}\) is greater than or equal to \(y^{2}\) in every component and greater in at least one component. Furthermore, we define the cone \(R^{m}_{\succeq }=\{x\in R^{m}: x\succeq 0\}\).
The Pareto efficient solution to (MLP) is defined as follows:
Definition 1
A feasible solution \(x^{*}\in X\) to (MLP) is Pareto efficient if there is no feasible solution \(x\in X\) such that \(f(x)\in f(x^{*})-R^{m}_{\succeq }\).
2.2 Robust multiobjective linear programs
Given an uncertain multiobjective linear programming problem (UMLP), there arises the same question of how to find feasible solutions \(x \in X\) as in a single objective optimization problem. We cannot take the worst case under all scenarios for evaluating solutions because in multiobjective optimization problems, we obtain that the objective value for each scenario is a vector. Therefore, let \(f_{U}(x)=\{f(x; \tilde{a}_{i},\tilde{b}_{i}):(\tilde{a}_{i},\tilde{b}_{i})\in U_{i}, i=1,2,\ldots, m\}\subset R^{m}\). We can generalize the concept of efficiency as given in Definition 1 with this notion.
Definition 2
A feasible solution \(x^{*}\in X\) to problem (UMLP) is robust efficient if there is no feasible solution \(x\in X\setminus \{x^{*}\}\) such that \(f_{U}(x)\subseteq f_{U}(x^{*})-R^{m}_{\succeq }\).
Thus, all possible objective values of a solution \(x^{*}\) are considered over all scenarios, namely the set \(f_{U}(x^{*})\).
Example 1
Having introduced the definition of robust efficient solutions for uncertain multiobjective linear programs (UMLPs), in Sections 3 and 4, we use two scalarization methods, the weighted sum method and the ϵ-constraint method, to find the robust efficient solution for (UMLP).
3 Weighted sum scalarization
Lemma 1
Given an uncertain multiobjective linear programming problem (UMLP), if \(x^{*}\in X\) is the unique optimal solution to (RWSP), then \(x^{*}\) is robust efficient solution for (UMLP).
Proof
We further show that robust efficient solutions for uncertain multiobjective linear programming problems with ellipsoidal uncertainty sets and general uncertainty sets can be found by solving deterministic optimization problems using weighted sum scalarization and thus can be computed by existing technology of deterministic optimization problems.
3.1 Ellipsoidal uncertainty sets
Theorem 1
Proof
3.2 General norm uncertainty sets
Theorem 2
Proof
Remark
- 1.
If the general norm uncertainty set is given by the Euclidean norm \(\Vert \cdot \Vert _{2}\), then (DP2) can be formulated as a second-order cone optimization problem because the Euclidean norm is self-dual.
- 2.
If the general norm uncertainty set is described by either \(\Vert \cdot \Vert _{1}\) or \(\Vert \cdot \Vert _{\infty }\), then (DP2) can be formulated as a linear programming problem.
- 3.We consider the uncertainty set described by the D-norm was studied by Bertsimas and Sim [4–6]. The D-norm of \(y\in R^{n}\) is given as follows:$$\Vert y \Vert _{p}=\max_{ \{S\cup {t}\vert S\subseteq N, \vert S\vert \le \lfloor p\rfloor,t\in N\setminus S \}} \biggl\{ \sum _{j\in S}\vert y_{j}\vert + \bigl(p- \lfloor p \rfloor \bigr)\vert y_{t}\vert \biggr\} , $$
By Theorem 2 we can obtain that if the general norm uncertainty set is given by the D-norm, then (DP2) can also be formulated as a linear programming problem.
4 ϵ-constraint scalarization
Lemma 2
Given an uncertain multiobjective linear programming problem (UMLP), if \(x^{*}\in X\) is the unique optimal solution to (RECP) for some \(\epsilon \in R^{m}\) and some \(t\in \{1,2,\ldots, m\}\), then \(x^{*}\) is a robust efficient solution for (UMLP).
Proof
Next, we show that robust efficient solutions for uncertain multiobjective linear programming problems with ellipsoidal uncertainty sets and general uncertainty sets can be found by solving deterministic optimization problems using ϵ-constraint scalarization and so can be computed by existing technology of deterministic optimization problems.
Theorem 3
Proof
Theorem 4
Proof
Remark
- 1.
If the general norm uncertainty set is given by the Euclidean norm \(\Vert \cdot \Vert _{2}\), then (DP4) can be formulated as a second-order cone programming with second-order cone constraints.
- 2.
If the general norm uncertainty set is given by either \(\Vert \cdot \Vert _{1}\) or \(\Vert \cdot \Vert _{\infty }\), then (DP4) can be formulated as a linear programming problem.
- 3.
If the general norm uncertainty set is described by the D-norm \(\Vert \cdot \Vert _{p}\), then (DP4) can also be formulated as a linear programming problem.
5 Conclusions
Multiobjective optimization and robust optimization have been well studied, but they are rarely considered in combination. In this paper, we consider the multiobjective linear programs where coefficients in the objective function belong to uncertain-but-bounded sets. First, we introduce the concept of a robust efficient solution to uncertain multiobjective linear programs. We also introduce two common scalarization methods, the weighted sum scalarization and ϵ-constraint scalarization, to compute robust efficient solutions of uncertain multiobjective linear programs. Finally, we obtain that the robust efficient solutions of uncertain multiobjective linear programs can be computed by some deterministic optimization problems using both weighted sum method and ϵ-constraint method.
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 11401484).
Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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