A new method for optimizing a linear function over the efficient set of a multiobjective integer program
Introduction
Many real world-problems involve multiple objectives. Due to conflict between objectives, finding a feasible solution that simultaneously optimizes all objectives is usually impossible. Thus, generating many or all efficient solutions, i.e., solutions in which it is impossible to improve the value of one objective without a deterioration in the value of at least one other objective, is usually the goal in multiobjective optimization.
We focus on multiobjective integer programs (MOIPs). Exact algorithms for MOIPs can be divided into decision space search algorithms, i.e., methods that search in the space of feasible solutions, and criterion space search algorithms, i.e., methods that search in the space of objective function values. During the last decade, several criterion space search methods have been developed for biobjective, triobjective, and general MOIPs, and, currently, they appear to be more effective than decision space search algorithms. Therefore, our focus will be on criterion search space methods.
Despite the recent progress in the development of exact algorithms for MOIPs, computing the efficient frontier is still not practical for many real-world problems. Furthermore, some researchers argue (see for instance Jorge, 2009) that presenting (too) many efficient solutions to a decision maker may, more than anything, confuse the decision maker and may make selecting a preferred solution almost impossible. An approach that alleviates this issue is finding the most desirable solution among the efficient solutions (preferably without enumerating all of them). This approach is known as optimizing over the efficient set, which is a global optimization problem (Benson, 1984).
Many researchers have addressed the problem of optimizing a linear function over the efficient set of a multiobjective linear program (MOLP). We refer the interested readers to the studies by Benson (1991, 1992, 1993, 1984), Dauer (1991), Ecker and Song (1994), Sayın (2000) and Yamamoto (2002). However, there are only a few studies that address the problem of optimizing a linear function over the efficient set of a MOIP. There are two main reasons for that:
- 1.
The set of (feasible) solutions to a MOLP, in the decision space as well as in the criterion space, is convex (assuming that the problem is feasible). Therefore, all efficient solutions to a MOLP are supported, i.e., they can be obtained by optimizing a weighted combinations of objective functions. Unfortunately, in general, this is not the case for a MOIP. As a consequence, determining the set of efficient solutions of a MOIP is (far) more challenging.
- 2.
The natural algorithm for optimizing a linear function over the efficient set of a MOIP, which enumerates the nondominated points of the MOIP and uses lower and upper bounds on the optimal value of the linear function to curtail the enumeration, requires the efficient enumeration of the nondominated points and the computation of effective bounds. Even though progress has been made on the former in recent years, little is known about how to accomplish the latter.
To the best of our knowledge, Abbas and Chaabane (2006) are the first to develop a method for optimizing a linear function over the efficient set of a MOIP. Their algorithm is a decision space search method. In each iteration, the method adds (different types of) cuts to guarantee an improvement in the value of the linear function. Unfortunately, Abbas and Chaabane do not report any computational results, which makes it difficult to assess the efficacy of their method.
Jorge (2009) is the first to develop a (simple) criterion space search algorithm. His approach is a variant of the method of Sylva and Crema (2004). Chaabane, Brahmi, and Ramdani (2012) and Chaabane and Pirlot (2010) present minor variations to Jorge’s algorithm. For instance, the algorithm in Chaabane et al. (2012) differs primarily in the scalarization technique used to determine whether a solution is efficient. Unfortunately, in both studies, no computational comparison with Jorge’s algorithm is included. As a result, it is not possible to determine whether the changes lead to improved performance.
A related problem, and a special cases of optimizing a linear function over the efficient set, is finding the nadir point, i.e., the point in criterion space given by the worst value of each of the objective functions over the set of efficient solutions. Recently, Köksalan and Lokman (2014) developed a new algorithm to find the nadir point for a MOIP. They showed that their approach is significantly faster than using Jorge’s algorithm. The main reason that their algorithm performs better than Jorge’s algorithm is that their algorithm decomposes the criterion space rather than adding more and more binary variables and disjunctive constraints to the single-objective IPs as is done in Jorge’s algorithm (and Sylva and Crema’s algorithm). Kirlik and Sayın (2014) also present an algorithm for finding the nadir point. Unfortunately, they did not compare their approach to the one of Köksalan and Lokman.
The main contribution of our research is the development of a new algorithm for enumerating the nondominated points of a MOIP and showing how the algorithm can be modified to optimize a linear function over the set of efficient solutions of a MOIP. The algorithm is an extension of the L-shape search method for triobjective integer programs (Boland, Charkhgard, & Savelsbergh, 2015) that works in the full-dimensional criterion space rather that a projected lower-dimensional criterion space. The algorithm has the following desirable characteristic: after finding a nondominated point, it adds at most one new subproblem to the list of subproblems still to be solved. As a consequence, the algorithm can be modified to efficiently optimize a linear function over the set of efficient solutions of a MOIP.
All criterion search space methods, except for the method of Sylva and Crema (2004), decompose the criterion search space into smaller subspaces (or subproblems) after finding a nondominated point; one of which can be deleted as it is dominated by the newly found nondominated point, but the others, which may still contain as-yet-unknown nondominated points, have to be searched at some future point in time (and possibly further decomposed). By maintaining a best known efficient solution, i.e., a known efficient solution with the best value of the linear function, and by computing a bound on the value of the linear function for a subspace, it may be possible to discard that subspace from further consideration. However, because computing such bounds is time consuming, the efficiency of such an approach depends strongly on the (total) number of subspaces created. The method of Sylva and Crema (2004) does not decompose the search space, but, instead, “eliminates” the dominated part of the search space by adding sets of disjunctive constraints (and associated binary variables) to the IP that searches for a nondominated point. As a result, searching for nondominated points becomes more and more time consuming, because the IPs get harder and harder to solve. This, of course, also implies that computing a bound on the value of the linear function becomes more and more time consuming.
By developing a new method for enumerating the nondominated points of a MOIP which, when decomposing the search space, never increases the number of subspaces that still need to be explored by more than one, and does not increase the complexity of the subspaces that still need to be explored, we are able to limit the time spent on computing bounds on the value of the linear function, because fewer bounds are computed (and, thus, there are fewer wasted bound computations), and each bound computation does not require the addition of a large number of sets of disjunctive constraints (and, thus, bound computations do not become more time consuming over time).
The proposed algorithm has two additional desirable properties: (1) because it maintains a lower and an upper bound on the value of the linear function at any point in time, it can be used to quickly generate a provably high-quality approximate solution, and (2) it can easily be modified to efficiently compute the nadir point of an MOIP.
An extensive computational study demonstrates that the proposed algorithm clearly outperforms Jorge’s algorithm (Jorge, 2009), the current state-of-the-art, on a variety of benchmark and randomly generated instances, and that it is competitive with customized algorithms for computing the nadir point.
The rest of paper is organized as follows. In Section 2, we introduce important concepts and notation. In Section 3, we present an introductory example. In Section 4, we detail the logic of the new algorithm. In Section 5, we review Jorge’s algorithm and show how this algorithm as well as our algorithm can be modified to compute the nadir point efficiently. In Section 6, we present the results of a comprehensive computational study. Finally, in Section 7, we give some concluding remarks.
Section snippets
Preliminaries
A multiobjective optimization problem can be stated as follows: where for all represents the feasible set in the decision space and the image of under vector-valued function represents the feasible set in the criterion space, i.e., for some . For convenience, we also use the notation for the nonnegative orthant of and for the positive orthant of .
When is
An introductory example
We first illustrate informally how the new algorithm computes the nondominated frontier of a MOIP on an instance with two objective functions. Suppose that the nondominated frontier contains the five points {z1, z2, z3, z4, z5} shown in Fig. 1a.
The idea underlying the new algorithm is to find, in each iteration, an as-yet-unknown nondominated point and to then decompose the criterion space. The algorithm maintains a priority queue of objects, each representing a (different) part of the
The algorithm
The algorithm maintains a priority queue of objects in the criterion space, each of which still has to be explored, i.e., may still contain as-yet-unknown nondominated points. Each object in the priority queue is characterized by a set of points defining its upper envelope, a point y to “guide” the search of the object, and a point yL defining its lower bound. The priority queue is initialized with the object defined by and . The algorithm also maintains a list of known
Optimizing a linear function over the set of efficient solutions
In this section, we start with a review of Jorge’s method (Jorge, 2009). We then show how the algorithm presented in the previous section can be modified to optimize a linear function over the set of efficient solutions. Finally, we show how these methods can be used to compute the nadir point.
A computational study
We have performed a computational study to assess the performance of the new algorithm for solving MOIPs with an arbitrary number of objective functions, and to assess the efficacy of the modified algorithm that can optimize a linear function over the efficient solution of a MOIP.
Our algorithms as well as Jorge’s algorithm have been implemented in C++ and use CPLEX 12.6 as the integer programming solver. Interested readers can find the sources of our algorithms at //hdl.handle.net/1959.13/1062187
Conclusion
We presented a new algorithm for optimizing a linear function over the set of efficient solutions of a MOIP. We have shown that it clearly outperforms Jorge’s algorithm (Jorge, 2009), which represents the current state-of-the-art. The algorithm’s success relies on the excellent performance a new algorithm for enumerating the nondominated points of a MOIP, which is achieved by employing a novel criterion space decomposition scheme which (1) limits the number of subspaces that have to be
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