The Quadrant Shrinking Method: A simple and efficient algorithm for solving tri-objective integer programs

https://doi.org/10.1016/j.ejor.2016.03.035Get rights and content

Highlights

  • We propose a new exact algorithm for solving tri-objective integer programs.

  • We prove that the algorithm solves a linearly bounded number of integer programs.

  • The algorithm is competitive with the state-of-the-art algorithms.

Abstract

We present a new variant of the full 2-split algorithm, the Quadrant Shrinking Method (QSM), for finding all nondominated points of a tri-objective integer program. The algorithm is easy to implement and solves at most 3|YN|+1 single-objective integer programs when computing the nondominated frontier, where YN is the set of all nondominated points. A computational study demonstrates the efficacy of QSM.

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. Consequently, in practice, decision makers want to explore and understand the trade off between objectives before choosing a suitable solution. 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 a primary goal in multiobjective optimization.

Exact algorithms for multiobjective integer programs (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. Criterion space search algorithms tend to be easier to implement and as effective, if not more effective, than decision space search algorithms. Therefore, we have chosen to focus on criterion space search algorithms. Readers interested in recent advances in the area of decision space search algorithms are referred to Belotti, Soylu, and Wiecek (2013), Stidsen, Andersen, and Dammann (2014), Vincent, Seipp, Ruzika, Przybylski, and Gandibleux (2013).

The simplest form of a multiobjective integer program (MOIP) has only two objectives and is known as a biobjective integer program (BOIP). The earliest algorithms only generated a subset of all nondominated points, the so-called extreme supported nondominated points, e.g., Aneja and Nair (1979). But not long after, algorithms were developed that generated the complete nondominated frontier. Presently, a number of effective and efficient criterion space search algorithms for solving BOIPs exist. Interested readers are referred to Ehrgott (2006) and Boland, Charkhgard, and Savelsbergh (2015a) for an overview of recent advances in solving BOIPs using criterion search space algorithms.

The situation is quite different when the number of objectives is greater than two; far fewer methods have been proposed for generating the complete nondominated frontier in these settings. The main reasons that generating the complete nondominated frontier of a MOIP with more than two objectives is significantly more challenging are that:

  • 1.

    The number of nondominated points tends to grow with the number of objective functions (see Brunsch, Goyal, Rademacher, and Röglin, 2014 for recent theoretical results that support this empirical observation). Since at least one search operation has to be performed to find a nondominated point, and searching for a nondominated point in (current) criterion space search methods involves solving a single-objective IP, the number of single-objective IPs that needs to be solved in order to solve even relatively small MOIP instances becomes quite large as the number of objectives grows.

  • 2.

    Developing effective strategies for decomposing the criterion search space is much more difficult, because such strategies need to balance the number and shape of the elements in the decomposition with the difficulty of the single-objective IP used to explore an element of the decomposition. Consequently, many single-objective IP solves or very expensive single-objective IP solves are unavoidable.

In this paper, we restrict ourselves to generating the complete nondominated frontier of a tri-objective integer program (TOIP).

Searching for an as-yet unknown nondominated point is a core operation in any method for generating the nondominated frontier of a TOIP and accounts for most of the solution time. It is done by solving one or more single-objective IPs depending on the scalarization technique used. As a consequence, the total number of IPs solved when generating the nondominated frontier of a TOIP depends on the scalarization technique used. The relevant scalarization techniques are (1) weighted-sum, in which one IP is solved which optimizes a weighted sum of the three objectives, and (2) lexicographic, in which a sequence of IPs is solved; in the case of a three-stage lexicographic scalarization, three IPs, each optimizing one of the objectives, or, in the case of a two-stage lexicographic scalarization, one IP optimizing one of the objectives and one IP optimizing a weighted sum of the other two objectives (or, possibly, of the three objectives). It is important to mention and emphasize that solving a single IP which optimizes a weighted-sum objective in the search for an as-yet unknown nondominated point (as in weighted-sum) is not necessarily more efficient than solving a sequence of IPs in the search for an as-yet unknown nondominated point (as in lexicographic). In fact, computational experiments reveals that it usually is not. The reason is that the IPs solved during a lexicographic search tend to be easier, and, other than the first, can be initialized with a high-quality known feasible solution. In the following, when reporting the number of IPs solved by a specific method, we assume, unless specifically stated otherwise, that the weighted-sum scalarization technique is used.

Sylva and Crema (2004) proposed a simple, straightforward algorithm for solving a TOIP, which solves exactly |YN|+1 single-objective IPs, where YN is the set of nondominated points (the algorithm solves one infeasible IP to determine that no other nondominated point exists). However, the single-objective IPs that need to be solved get harder and harder (as more and more disjunctive constraints are added), and, as a consequence, in practice, after only a few iterations, solving the single-objective IPs becomes computationally prohibitive.

More sophisticated algorithms solve more, but easier IPs (compared to the algorithm of Sylva and Crema) and end up being more efficient in practice, including the enhanced recursive method (Özlen, Burton, & MacRae, 2013), the full 3-split method (Dächert, Klamroth, 2014, Dhaenens, Lemesre, Talbi, 2010), the full 2-split method (Dächert, Klamroth, 2014, Kirlik, Sayın, 2014, Lokman, Köksalan, 2013), and the L-shape search method (Boland, Charkhgard, & Savelsbergh, 2015b). (We note that besides Dächert and Klamroth (2014) and Boland et al. (2015b), all cited methods are designed for any number of criteria, not only for TOIPs.)

A full 3-split method works with 3-dimensional boxes in criterion space that may still contain as-yet unknown nondominated points. It searches a box to find a new nondominated point. When a new nondominated point is found, the box is decomposed into smaller boxes and the process is repeated in each of the new, smaller boxes. It can be shown that an implementation of the method exists that solves at most 3|YN|+1 IPs (Dächert & Klamroth, 2014).

The full 2-split method is similar to the full 3-split method. The difference is that the full 2-split method works in a projected (2-dimensional) space rather than the original (3-dimensional) criterion space. The main advantage of doing so, in practice, is that the number of infeasible IPs that has to be solved is much smaller, and single-objective IP solvers tend to struggle more when solving infeasible IPs. Because the full 2-split method operates in a 2-dimensional space, it works with rectangles instead of boxes. Depending on the variant of the method, the number of IPs solved is bounded by O(|YN|2) (Kirlik & Sayın, 2014) or 2|YN|+1 IPs (Dächert & Klamroth, 2014) (or 3|YN|+1 IPs if a two-stage scalarization technique is used).

The L-shape search method (LSM) combines ideas of the algorithm of Sylva and Crema and the full 2-split method. It also operates in a projected space. However, instead of working only with rectangles, it also works with L-shapes. As far as we know, LSM is the most efficient algorithm to compute the (complete) nondominated frontier of a TOIP. Another important feature of LSM is that it can quickly produce a high-quality approximate nondominated frontier. It can be shown that the number of IPs solved by LSM is bounded by O(|YN|2).

In this paper, we present a simple, easy to implement, but very efficient variant of the full 2-split method, the Quadrant Shrinking Method (QSM). In addition, we show that QSM solves at most 3|YN|+1 single-objective IPs (QSM uses two-stage scalarization). This bound equals the bound obtained (independently) by Dächert and Klamroth (2014) for their variant of the full 2-split method. QSM solves at most |YN|+1 infeasible IPs, which may be one reason for its excellent performance. Furthermore, |YN| of the single-objective IPs solved by QSM convert a weakly nondominated point into a nondominated point, which typically can be done extremely fast, because a near-optimal feasible solution can be provided to jump-start the search.

To demonstrate the efficacy of QSM, a computational study has been conducted using publicly available instances of the tri-objective one-dimensional knapsack problem (1DKP), the tri-objective three-dimensional knapsack problem (3DKP), the tri-objective assignment problem (AP), and the tri-objective traveling salesman problem(TSP), in which we compare the performance of QSM to the performance of LSM, KSA (the variant of the full 2-split method developed by Kirlik & Sayın, 2014), ERM (the enhanced recursive method developed by Özlen et al., 2013), and DKA (a reimplementation of the full 2-split method developed by Dächert & Klamroth, 2014). The results show that QSM is significantly faster than any of the existing methods.

The superior performance of QSM, compared to existing methods, can be attributed to its novel decomposition technique (which immediately detects redundancy of many of the constructed rectangles), its simple data structures, its seeding of single-objective IPs with high-quality feasible solutions, and its use of enhancements aimed at eliminating redundant calculations.

The rest of paper is organized as follows. In Section 2, we introduce critical concepts of tri-objective integer programming, detail the logic of QSM, and prove the bound on the number of single-objective IPs solved by QSM. In Section 3, we discuss the results of our computational study. Finally, in Section 4, we give some concluding remarks.

Section snippets

The Quadrant Shrinking Method

A TOIP can be stated as follows minxXz(x)=(z1(x),z2(x),z3(x))where XZn is defined by a set of affine constraints that represents the feasible set in the decision space and z1(x), z2(x) and z3(x) are linear functions. The image Y of X under the vector-valued function z:ZnR3 represents the feasible set in the criterion space, i.e., Y:=z(X):={yR3:y=z(x) for some xX}. For convenience, we also define the following sets: R3:={yR3:y0}, the nonnegative orthant of R3, and R>3:={yR3:y>0}, the

A computational study

QSM is implemented in C++ and uses CPLEX 12.6 as the integer programming solver. The computational experiments are conducted on a computer with 2.30 gigahertz Intel Xeon E5-2650 processors and 264 gigabyte RAM, with the RedHat Enterprise Linux 7.1 operating system, and using a single thread.

In our computational study, we use two publicly available sets of instances, Set I and Set II, used in previous studies of methods for solving TOIPs. The first set has been used by Özlen et al. (2013) and

Conclusion

We have introduced an efficient (and easy to implement) variant of the full 2-split method for TOIPs. We have shown that QSM solves at most 3|YN|+1 single-objective IPs. Moreover, computational experiments demonstrate that QSM outperforms existing algorithms for TOIPs.

References (15)

There are more references available in the full text version of this article.

Cited by (50)

  • Optimization of a quadratic programming problem over an integer efficient set

    2024, Journal of Computational and Applied Mathematics
View all citing articles on Scopus
View full text