Integrating column generation in a method to compute a discrete representation of the nondominated set of multiobjective linear programmes
Abstract
In this paper we propose the integration of column generation in the revised normal boundary intersection (RNBI) approach to compute a representative set of nondominated points for multiobjective linear programmes (MOLPs). The RNBI approach solves single objective linear programmes, the RNBI subproblems, to project a set of evenly distributed reference points to the nondominated set of an MOLP. We solve each RNBI subproblem using column generation, which moves the current point in objective space of the MOLP towards the nondominated set. Since RNBI subproblems may be infeasible, we attempt to detect this infeasibility early. First, a reference point bounding method is proposed to eliminate reference points that lead to infeasible RNBI subproblems. Furthermore, different initialisation approaches for column generation are implemented, including Farkas pricing. We investigate the quality of the representation obtained. To demonstrate the efficacy of the proposed approach, we apply it to an MOLP arising in radiotherapy treatment design. In contrast to conventional optimisation approaches, treatment design using column generation provides deliverable treatment plans, avoiding a segmentation step which deteriorates treatment quality. As a result total monitor units is considerably reduced. We also note that reference point bounding dramatically reduces the number of RNBI subproblems that need to be solved.
Keywords
Multiobjective linear programming Column generation Revised normal boundary intersection method Radiotherapy treatment designMathematics Subject Classification
90C29 90C05 90C901 Introduction
Multiobjective optimisation (MOO) deals with optimisation problems involving several conflicting objectives. In MOO, a single solution that simultaneously optimises all objectives generally does not exist. Instead, MOO seeks for solutions that cannot improve in any single objective without deteriorating at least one other objective. Solutions with this property are referred to as efficient solutions. The points obtained by mapping the efficient solutions to the objective space are referred to as nondominated points. The purpose of MOO is to obtain the nondominated set and one efficient solution in the preimage of every nondominated point. A decision maker then has the task to select the most preferred nondominated point and a corresponding efficient solution for the problem at hand. In multiobjective continuous optimisation, the nondominated set consists of infinitely many nondominated points. It is therefore impractical for a decision maker to examine all nondominated points. Instead, a practical approach is to obtain a discrete representation of the nondominated set satisfying some quality requirements (Sayın 2000; Faulkenberg and Wiecek 2010). Many methods that follow this approach have been proposed in the last two decades, as the paper by Faulkenberg and Wiecek (2010) shows. Given this representative nondominated set, the decision maker can navigate through the nondominated points and decide on the most preferred point. In this study we propose to integrate column generation in an approach to find a representative nondominated set for multiobjective linear programmes (MOLPs).
Column generation is a technique that solves linear programmes by considering only a subset of the decision variables. The technique is particularly beneficial when the number of variables is much greater than the number of constraints. The idea is based on the fact that, typically, only a subset of variables is required in the basis to reach optimality; other variables are nonbasic and have a value of zero. Column generation exploits this fact by only considering variables that have the potential to improve the objective function value, indicated by negative reduced costs. In each iteration of a column generation method, two problems need to be solved successively: the restricted master problem (RMP) and the subproblem (SP). RMP is the original problem with only a subset of variables. By solving the RMP, a vector of dual values associated with the constraints of the RMP is obtained. The dual information is passed on to the SP. The goal of the SP is to identify a new variable and an associated coefficient column with negative reduced cost, which can potentially improve the objective function value of the original problem. If such a variable and column can be identified, then they are added to RMP, which is reoptimised, and the next iteration begins. Otherwise, an optimal solution of RMP is also an optimal solution of the original problem.
Column generation methods in multiobjective optimisation are rare. Moradi et al. (2015) present a column generation approach for the (linear) biobjective multicommodity minimum cost flow problem. Their algorithm incorporates column generation within a biobjective simplex algorithm, which requires a modification of the objective function of the SP to a linear fractional function. The study of Salari and Unkelbach (2013) falls into the domain of nonlinear programming, thus the subproblem is based on partial derivatives of individual objective functions. The aim of Salari and Unkelbach (2013) is to approximate the entire nondominated set using a limited number of variables. The basic idea is to use column generation to identify variables that potentially improve the nondominated set approximation as a whole. To find such variables, multiple weightedsum RMPs, where each RMP is associated with a unique nonnegative weight vector, are solved. The partial derivatives obtained from solving each RMP are passed to a subproblem, which aggregates the individual subproblems corresponding to each RMP. A column obtained from solving the aggregated subproblem therefore potentially improves the majority of individual RMPs, thus improving the nondominated set approximation as a whole. However, due to the use of weight vectors for the RMPs and the use of an aggregated subproblem, their method cannot guarantee that the whole nondominated set is well approximated.
In this study, we propose to use column generation within a procedure that constructs an evenly distributed finite representative set of nondominated points of an MOLP, i.e. the revised normal boundary intersection (RNBI) method of Shao and Ehrgott (2007, 2016). The RNBI method combines aspects of the global shooting method (Benson and Sayın 1997) and the normal boundary intersection method (Das and Dennis 1998) and has been proven to generate evenly distributed nondominated points for MOLPs (Shao and Ehrgott 2007). Unlike the method of Salari and Unkelbach (2013) in which a subproblem identifies a variable that improves the nondominated set approximation in general, each of the column generation subproblems in our approach identifies a variable and an associated column to move a point in objective space in a direction that leads to nondominance. In fact, if column generation is run to termination, i.e. optimality of the master problem, the resulting point will be on the boundary of the feasible set of the MOLP in objective space.
We apply our method to a multiobjective optimisation problem in radiotherapy treatment design. The goal of this problem is to identify a treatment plan (in the form of socalled fluence maps for several radiation beams) in order to deliver a tumouricidal dose of radiation to a planning target volume, while sparing healthy tissue. These conflicting goals naturally lead to formulations as multiobjective optimisation problems. We refer the reader to Ehrgott et al. (2008a) for more details on optimisation methods in radiation oncology. By applying our column generation RNBI method to the treatment design problem, a set of representative treatment plans, each with a unique tradeoff between objective function values, are generated. Given these plans, the oncologist can then decide on the plan that best benefits the patient.
Conventional and multiobjective approaches in radiotherapy treatment design generate treatment plans that often cannot be practically delivered by existing radiotherapy equipment. In order to make them deliverable, one needs to modify the treatment plans to incorporate physical delivery constraints and to reduce the total time a patient is exposed to radiation. This modification, which is referred to as segmentation, deteriorates treatment plan quality (Rocha et al. 2012; Craft and Richter 2013). Thus, should a treatment plan become unsatisfactory after segmentation, the treatment planner will have to reoptimise and find another plan. This iterative process makes the treatment design process inefficient. However, we shall see that the fluence map optimisation problem can be reformulated via decomposition to include the physical delivery constraints. It can then be solved by column generation to obtain treatment plans that are directly deliverable. In fact, we shall see that column generation produces plans that are close to optimality with a reduced delivery complexity, which are often preferable to optimal (but complex) plans in practice (Carlsson and Forsgren 2014; Broderick et al. 2009).
In Sect. 2, we provide background and formulations of single objective column generation and the RNBI method. In Sect. 3, we introduce the column generation RNBI formulation and discuss implementation issues associated with the method, i.e. the detection of infeasibility through a reference point bounding method and initialisation of the process. The quality of the representative set obtained by our column generation based RNBI method is also discussed. In Sect. 4, we apply the method to a prostate radiotherapy treatment planning problem, followed by results and discussion in Sect. 5.
2 Column generation and the RNBI method
In this section we provide necessary details for column generation and the RNBI method. For further details of these two topics, we refer the readers to Lübbecke (2010) and Shao and Ehrgott (2007), respectively.
2.1 Column generation
2.2 The RNBI method
Definition 1
A feasible solution \(\hat{x} \in X\) of MOLP is called efficient if there does not exist any \(x\in X\) such that \(c^k x \leqq c^k \hat{x}\) for \(k=1,\ldots ,p\), and \(Cx\ne C \hat{x}\). The set of all efficient solutions of the MOLP is called the efficient set \(X_E\) in decision space. The image in objective space \(\hat{y}=C \hat{x}\) of efficient solution \({\hat{x}}\) is called nondominated point. The set of all nondominated points is referred to as the nondominated set \(Y_N\) in objective space.
2.2.1 Constructing the reference subsimplex and choosing reference points
2.2.2 Computing the intersection points and checking nondominance

RNBISub is infeasible if and only if the halfline \(\{q+te: t \geqq 0\}\) does not intersect Y.

RNBISub has an optimal solution \(t^*\), but the intersection point \(q+t^*e\) of the halfline \(\{q+te: t \geqq 0\}\) and Y is dominated.

RNBISub has an optimal solution \(t^*\) and \(q+t^*e\) is a nondominated point of Y.
3 The RNBI method using column generation
Notice that RMPRNBISub is essentially the same as the RNBI subproblem but with only a subset of variables \(j \in J'\). To conduct column generation on this RNBI subproblem, we solve RMPRNBISub and the corresponding SP sequentially and iteratively. We remark that, in case column generation is terminated early, i.e. an optimal solution of RNBISub is not yet confirmed, the intersection point may be dominated. In contrast to the original RNBI method, nondominance of the intersection points will not be checked, because only an optimal solution of RNBISub can define a nondominated point.
As indicated in Sect. 2.2.2 RNBISub may be infeasible even in the presence of all variables. Hence, if we solve RMPRNBISub with a subset of variables, it may be infeasible because either the constraints (7c) are not satisfied with a subset of variables or because the master problem RNBISub is infeasible, i.e. \(\{q+te: t \geqq 0\}\) does not intersect Y. The former case can be dealt with by the use of artificial variables to satisfy constraints (7c), see also Sect. 3.1. But in the latter case, many iterations of column generation may be wasted to detect the infeasibility. In fact, infeasibility of RNBISub could only be determined once all artificial variables are eliminated from the solution.
It will therefore be beneficial to identify reference points for which this is the case early to avoid attempts to solve RNBISub for such reference points. For convenience, we will from now on refer to reference points for which RNBISub is infeasible as infeasible reference points. In Sect. 3.1 we present a method, which we call reference point bounding, to identify infeasible reference points. To deal with infeasibility due to the restricted number of variables in RMPRNBISub, we present three methods of initialisation in Sect. 3.2. Finally, we discuss the quality of the representation generated by column generation RNBI in Sect. 3.3.
3.1 Reference point bounding
One issue with the RNBI method, which stems from the use of the antiideal point in the definition of the covering simplex S and the reference subsimplex \({\hat{S}}\), is that there can be infeasible reference points, i.e. reference points for which RNBISub is infeasible such that \(\{q+te: t \geqq 0\} \cap Y = \emptyset .\) Because the components of \(y^{AI}\) may be far larger than the objective values of any nondominated point, there can potentially be many reference points for which this is also the case, as shown in Fig. 1. Obviously, any effort invested in solving RNBISub for infeasible reference points is wasted in the sense that it does not contribute to the computation of a representative set of nondominated points. Therefore, solving RNBISub using column generation if RNBISub is in fact infeasible, can dramatically increase the computational time (see Sect. 4.2). In order to identify infeasible reference points we provide Theorem 1 characterising infeasible reference points and therefore defining the subset of feasible reference points of \({\hat{S}}\). We first state a lemma concerning the set of all feasible reference points.
Lemma 1
The subset \({\hat{Q}} \subset {\hat{S}}\) of points q such that \(\{q+te: t\geqq 0\} \cap Y \ne \emptyset \) is a polytope.
Proof
The result follows, because \({\hat{Q}}\) is the projection of polytope Y onto \({\hat{S}}\), which is a simplex on the hyperplane \(e^Ty= \mu \). \(\square \)
Theorem 1
Let \(q \in {\hat{S}}\) be a reference point. Then q is infeasible if and only if there is some \(d \in {\mathbb {R}}^p \setminus \{0\}\) such that \(d^Tq < \min \{d^Tz: z+te \in Y, z \in {\hat{S}}, t \geqq 0\}.\)
Proof
We first observe that the LP \(\min \{d^Tz: z+te \in Y, z \in {\hat{S}}, t \geqq 0\}\) is always feasible, because \(z^* = {\hat{y}}, t^*=0\) with \({\hat{y}}\) as defined in Sect. 2.2.1 is a feasible solution. It is also bounded, because Y is a compact set by assumption. Then \(d^Tq < \min \{d^Tz: z+te \in Y, z \in {\hat{S}}, t \geqq 0\}\) implies that q does not satisfy \(q+te \in Y\) for any \(t \geqq 0\). Now let q be an infeasible reference point. Then \(q \notin {\hat{Q}}\) as defined in Lemma 1. Hence there exists a hyperplane strictly separating q from \({\hat{Q}}\), i.e. there is \(d \in {\mathbb {R}}^p \setminus \{0\}\) such that \(d^Tq < \min \{d^Tz: z \in {\hat{Q}}\} = \min \{d^Tz: z+ty \in Y, z \in {\hat{S}}, t \geqq 0\}.\) \(\square \)
Although Theorem 1 provides a theoretical characterisation of all feasible reference points, it is clearly impractical for implementation. Hence, we restrict ourselves to finding minimum and maximum values of each individual coordinate \(z_k\) of points on the reference subsimplex \({\hat{S}}\) that are feasible reference points, i.e. we use the sufficient condition of Theorem 1 and apply it to vectors \(d=e^k\) and \(d=e^k\) for \(k=1, \ldots , p,\) where \(e^k\) is the kth unit vector. We call this method reference point bounding.
The linear programme \(\min \{d^Tz: z+te \in Y, z \in {\hat{S}}, t \geqq 0\}\) is solved for \(d = e^k\) and \(d=e^k\) for \(k=1,\ldots ,p\). Let the optimal values be \(z_k^{min}\) and \(z_k^{max}\), respectively. Then according to Theorem 1, reference points q with \(q_k < z_k^{min}\) or \(q_k >z_k^{max}\) for any \(k \in \{1, \ldots , p\}\) will be infeasible. Corollary 1 summarises the above argument.
Corollary 1
If q is a reference point with \(q_k<z_k^{min}\) or \(q_k>z_k^{max}\) for some \(k\in \{ 1,\ldots ,p \}\), then \(\{q+te: t \geqq 0\} \cap Y = \emptyset \).
3.2 Initialisation of RMPRNBISub
Constraints (7b) may not be feasible given a limited set of variables. In addition, even after the reference point bounding procedure is applied, infeasible reference points may remain due to \(\{q+te: t \geqq 0\} \cap Y =\emptyset \). In this section we discuss how the infeasibility of RMPRNBISub can be managed.
One way to handle the infeasibility is the Phase1 approach, see e.g. Chvátal (1983), which adds nonnegative artificial variables to satisfy constraints (7b) and (7c) while changing the objective function of the problem to minimise the sum of the artificial variables. The BigM approach assigns large costs M to the artificial variables and minimises the sum of the original objective function plus the sum of the costed artificial variables. Using artificial variables, feasibility of RMPRNBISub is assured. As soon as any of the artificial variables has a value of zero in a solution, the artificial variable can be removed. If any of the artificial variables remain positive when the optimality condition is satisfied, we can conclude that RMPRNBISub is infeasible because \(\{q+te: t \geqq 0\} \cap Y =\emptyset \).
We notice that in practice, column generation is rarely used to solve a (single objective) linear programme to optimality. In this situation, a possible approach is to perform column generation iterations on RMPRNBISub until a specified termination condition, such as a prespecified number of columns, is reached. One can, for example, conclude that a reference point is (approximately) feasible, if the solution satisfies constraints (7c) and the remaining total infeasibility in constraints (7b) is small enough, i.e. below a certain predetermined threshold.
An alternative approach to manage infeasibility is to generate coefficient columns that show that the RMP is feasible (Andersen 2001). The method is based on Farkas’ lemma, which states that either \(Ax=b, x \geqq 0\) is feasible or there is a vector \(\pi \) with \(\pi ^T A \geqq 0\) and \(\pi ^T b<0\). The vector \(\pi \) corresponds to the dual vector of a linear programme. A linear programme is proved to be infeasible by finding a dual vector such that the condition \(\pi ^T Ax = \pi ^T b\) can never be met due to opposite signs on the righthand side and the lefthand side of the equation. Thus to prove that the restricted master problem is feasible, we can add a column a to A with \(\pi ^T a \leqq 0\). Such a column can be found by solving \(\min {\{\pi ^T a( \lambda ): \pi ^T a( \lambda ) \leqq 0, \lambda \in X_J\} }\). If no such column exists, we can conclude the corresponding master problem is infeasible. We will refer to this approach as Farkas pricing.
3.3 Quality of the representative set computed by the column generation RNBI method
Sayın (2000) defines three measures, coverage, uniformity and cardinality, to quantify the quality of a discrete representation of a set. A good representation of the nondominated set should not contain an excessive number of points (low cardinality), should have points significantly different from one another (as indicated by high uniformity level) and should not neglect large portions of the nondominated set (low coverage error).
Let \(G \subset Y_N\) be a finite set of nondominated points generated by the standard RNBI method using reference points \(q \in Q\). Let H be the representative set generated by the RNBI method using column generation based on the same set of reference points. We shall write g(q) and h(q), respectively, to indicate the dependence of representative points on reference point q. The distance between two adjacent reference points is denoted as dq. Cardinality represents the number of points contained in the representation. It is clear that the number of points contained in H depends on the distance between adjacent reference points. In the rest of this section we discuss the quality of H in terms of uniformity level and coverage error.
Based on the above discussion, we can see that the quality of a representation generated by column generation RNBI depends on the distance between adjacent reference points. As the distance decreases, cardinality increases, the uniformity level decreases and the coverage error decreases. In addition, the coverage error also depends on the maximum distance between representative points g(q) and h(q) for reference points \(q \in Q\), which depends on the termination condition of the column generation process. Consequently, given a problem at hand, one should select a dq and a column generation termination condition that results in appropriate uniformity level and coverage level for the representative nondominated set.
4 Application of the column generation RNBI method in radiotherapy treatment design
From now on, we assume that beam directions are given and refer to Ehrgott et al. (2008c) for an overview of the problem of determining beam directions. Hence, given a set of beam directions, we are interested in finding a design, consisting of a set of segments and the associated radiation intensities, which best benefits the patient. Conventionally, the radiotherapy treatment design problem is split into two sequential optimisation problems, the fluence map optimisation (FMO) problem and the segmentation problem. FMO is the problem of finding the optimal modulated radiation intensity for each beam direction. To model the FMO problem mathematically, the radiation field at a beam direction is discretised into small sized rectangular subfields called bixels. These correspond to the smallest openings of the MLC, i.e. are of the width of one of the leafs of the MLC and of the length that corresponds to the distance between two stop positions of the leaf. FMO finds the radiation intensity for each bixel such that a desirable dose distribution that meets the goals of the treatment can be delivered to the patient. The intensities for the bixels are referred to as the intensity pattern. In principle, one can deliver the intensity pattern bixelbybixel using bixelsized segments. However, doing so would lead to an unrealistically long treatment time. In practice, the intensity pattern is realised by stacking a limited number of shaped radiation fields, each passing through an associated segment. Therefore, after obtaining the intensity pattern as output from solving the FMO problem, it is necessary to solve a socalled segmentation problem, which finds a set of segments that best realise the intensity pattern by, for instance, minimising the total beamon time required to deliver the intensity pattern or by minimising the required number of segments (see Baatar et al. 2005).
The segmentation problem is an optimisation problem that needs to incorporate physical constraints of the MLC leaves. The elementary ones are collision constraints, that prevent opposing leaves to overlap and constraints that ensure the opening in any MLC row is continuous, i.e. all open bixels in a row are consecutive. Other constraints are specific to particular brands of MLCs, which is why we concentrate on the basic ones in this study. To avoid generating overly complex treatment plans that cannot be practically delivered, the bixel intensities are discretised into a range of intensity levels at the beginning of the segmentation process. As a result, the intensity pattern is realised approximately and the quality of the treatment plan deteriorates after segmentation. A survey of the literature on segmentation problems can be found in Ehrgott et al. (2008b).
FMO needs to deal with several conflicting objectives associated with the PTV and the surrounding structures. The conflicting objectives in FMO have conventionally been handled by scalarisation (see Ehrgott et al. 2008a for a review). However, using this approach, if the generated intensity pattern is not satisfactory, the planner will have to iteratively adjust the plan optimisation parameters and reoptimise until a satisfactory intensity pattern is found. This process is time consuming and without guarantee of finding the best possible intensity pattern under the patient specific conditions. Instead, multiobjective optimisation has been introduced to solve the FMO problem. By generating a representative set of nondominated plans, the planner can browse the plans and choose the best one available without the iterative process. Several approaches have been proposed to solve multiobjective FMO problems, including goal programming methods (Falkinger et al. 2012; Breedveld et al. 2009; Wilkens et al. 2007; Jee et al. 2007), constraint methods (Hoffmann et al. 2006; Craft et al. 2005; Küfer et al. 2003; Hamacher and Küfer 2002) and approximation methods (Shao and Ehrgott 2008; Craft et al. 2006). In addition, RNBI has also been applied to multiobjective FMO problems by Shao and Ehrgott (2007).
Recently, a multiobjective FMO optimisation approach has been deployed in clinical practice (Craft and Richter 2013). The approach approximates the nondominated set using convex combinations of efficient solutions. However, since FMO does not consider plan delivery, the treatment plan generated from FMO needs to go through the segmentation process, which transforms an optimal intensity pattern into a limited number of segment intensities, thereby deteriorating plan quality, as demonstrated by Rocha et al. (2012). If the deliverable plan is not satisfactory, the planner will have to reoptimise and find another plan. To avoid this drawback, Craft and Richter (2013), Salari and Unkelbach (2013) and Fredriksson and Bokrantz (2013) have proposed multiobjective approaches to find deliverable plans. These approaches use convex combinations of the segmented plans or conical combinations of the segments to approximate the feasible set of the FMO problem and then use multiobjective interactive optimisation methods to navigate among the nondominated set of the approximated feasible set.
The clinically adopted MOO method uses a sandwiching method (see, e.g., Rennen et al. 2011; Bokrantz and Forsgren 2013) to generate an approximation of the nondominated set, followed by plan navigation on the convex hull of a set of plans (Monz et al. 2008). While approximating the nondominated set based on interpolation of a set of existing plans can reduce the computational expense, compared to generating a discrete representative nondominated set, we note that the interpolated solution may be subject to further improvement potential due to approximation error (see, e.g., Bokrantz and Miettinen 2015). In contrast, the RNBI procedure produces a discrete set of efficient plans that captures all potential treatment tradeoffs. Moreover, it is guaranteed that each nondominated point is no further than a known coverage error from the objective vector of one of the computed plans. Since it is also guaranteed that the objective vectors of the computed plans cover the entire nondominated set (in terms of a guaranteed minimum distance, called the uniformity level, between any two of them) there is no need to consider convex combinations of plans. As interpolation is not used to form plans, each plan is given freedom in beam angle configuration, segment shapes and segment intensities and hence allows one to achieve the bestquality plans for different treatment tradeoffs. In addition, since a set of discrete plans that captures different treatment tradeoffs are generated, navigation can be conducted by examining the existing set of plans. Consequently, one can extract relevant (nonconvex) clinical evaluation criteria, e.g., the dosevolume parameters and treatment delivery time, from the plans and use these criteria to find the most preferable plan from the representative nondominated set (Lin and Ehrgott 2016).
Column generation has been used to generate deliverable plans for single objective radiotherapy plan optimisation (PreciadoWalters et al. 2004; Romeijn et al. 2005). Here, the physical delivery constraints in the segmentation process are considered in the column generation subproblem. Essentially, each column generated from the subproblem represents a segment that is likely to improve the objective function value. As a result, the solutions produced from column generation can be delivered without additional segmentation. As will be demonstrated by our results, column generation produces plans that are nearoptimal and can be delivered with dramatically lower total monitor units than the corresponding optimal ones followed by segmentation. Such plans are desirable as they can be delivered with a shorter treatment time, i.e., exposing patients to radiation for a shorter time, and with lower radiation leakage from the MLCs (Broderick et al. 2009). In fact, nearoptimal plans that can be delivered efficiently and accurately are often preferable to complex optimal solutions in practice (Carlsson and Forsgren 2014). Earlier studies on deliverable multiobjective optimisation in radiotherapy limit the number of segments requiring that all computed plans use either the same segments (Salari and Unkelbach 2013), that the number of segments in each plan is limited (Craft and Richter 2013), or that the plans use both a subset of segments from a common pool and some individual ones (Fredriksson and Bokrantz 2013). We use column generation within the RNBI method to control the number of segments that are generated for each plan, making no restrictions on the set from which these segments are drawn. Because the method provides quality guarantees for the computed plans, the planner is then able to decide how many segments to allow solely based on plan quality without setting any apriori limits.
In this paper, we apply column generation within the RNBI framework for multiobjective radiotherapy treatment design. This approach produces a representative set of plans that are deliverable and are close to efficient plans that do not consider the deliverability of intensity patterns. The approach therefore combines the advantage of considering multiple objectives in the FMO problem with the advantage of producing deliverable intensity maps without much deterioration of treatment quality, which column generation delivers.
4.1 Formulation
4.2 The test case
We apply both the original RNBI method and the column generation RNBI method to a prostate radiotherapy treatment design problem. The RNBI method solves the RNBISub reformulation of (12) with bixel intensity variables x, producing a set of (not necessarily deliverable) intensity patterns that define a representative set of nondominated points for MOLP (12). The column generation RNBI method solves the RMPRNBISub reformulation of (12) with a subset of segment intensity variables \(\bar{x}\). We consider three objective functions: one for the PTV (objective 1), one for the rectum (objective 2) and one for the bladder (objective 3). Other clinically relevant structures such as the prostate, the right and left femural head and normal tissues, are involved in the formulation as constraints, e.g. voxels of the prostate are given a lower bound and an upper bound on the delivered dose and voxels of femural heads and normal tissues are given structure specific upper bounds. By only involving three objective functions, we are able to illustrate the results graphically.

no variable with a negative reduced cost can be found

the number of segments assigned with a positive intensity in a solution exceeds 100

the number of column generation iterations (or equivalently the number of segments) exceeds 150 .
4.3 Results
For convenience, representative points generated using RNBI and column generation RNBI will be referred to as RNBI points and the CGRNBI points, respectively. The representative sets of the CGRNBI points, grouped according to the number of positive segments, will be denoted as CGnumber with number being the corresponding number of positive segments.
Objective values and average computation time (rounded to seconds) of RNBISub and CGRNBISub with 40 positive segments (CG40) and 100 positive segments (CG100)
Reference point  Standard  CG40  CG100 

1  1.7450  1.9880  1.8140 
2  2.6806  2.9661  2.7525 
3  0.4216  0.7765  0.5019 
4  1.7451  2.0407  1.8215 
5  4.0186  4.2616  4.0875 
6  2.6941  2.9738  2.7695 
7  1.5498  1.9892  1.7157 
8  1.5417  1.9638  1.7104 
9  1.8497  2.2986  1.9810 
10  4.0186  4.2959  4.0916 
11  3.8230  4.2212  3.9890 
12  3.8108  4.1933  3.9770 
13  3.8108  4.2176  3.9943 
14  3.8112  4.2856  3.9865 
15  3.8153  4.3023  3.9846 
16  6.0843  6.4450  6.2286 
17  6.0843  6.4846  6.2481 
Average time  17  43  553 
The objective function values and the average computation time of the RNBI points, CG40 and CG100 are shown in Table 1. The objective function values of the points in CG40 and CG100 are on average 0.3647 and 0.1264 Gray higher than the objective function values of the RNBI points. The average computation time used to obtain the RNBI points, CG40 and CG100 are approximately 17, 43 and 553 seconds, respectively. We observe that, as the number of generated columns increases, the computation effort for solving RMPRNBISub increases as well.
Total monitor units for delivering the intensity patterns generated by RNBI and CGRNBI with 40 positive segments (CG40) and 100 positive segments (CG100)
Reference point  Standard  CG40  CG100 

1  494  97.8  155.4 
2  474  82.2  146.6 
3  444  79.1  138.0 
4  493  97.2  152.3 
5  494  97.8  155.4 
6  469  72.9  136.8 
7  341  66.4  119.7 
8  354  70.0  116.3 
9  446  79.7  135.6 
10  493  87.1  144.8 
11  343  65.8  103.6 
12  350  67.2  94.6 
13  350  60.8  106.5 
14  351  68.9  95.9 
15  354  77.9  108.0 
16  350  65.7  98.8 
17  350  66.7  96.3 
Averge MUs  408.8  76.7  123.8 
We use (8) to measure the uniformity of the representative sets. The results show that the uniformity levels for all representative nondominated sets are the same up to 4 decimal places, with a value of 3.2153 Gray, which is the same as the distance between any two closest reference points. However, the two closest intersection points that define the uniformity level are different for the different representative sets.
Minimum and maximum value for each objective based on the reference point bounding solved to optimality and solved by column generation (CG)
Optimal minimum  Optimal maximum  CG minimum  CG maximum  

Objective 1  \(\)7.5352  3.9529  \(\)7.2641  3.5035 
Objective 2  \(\)4.4038  9.159  \(\)4.0101  8.9119 
Objective 3  \(\)4.5379  9.1226  \(\)4.2902  8.8587 
Number of iterations required for Farkas pricing to identify RNBISub infeasibility
Iterations used to identify infeasibility  2  3  4  5  10  \(>150\) 
Number of RNBISub instances  48  10  1  6  1  8 
We also test the performance of Farkas pricing in concluding the infeasibility of RNBISub instances. Note that we have 91 reference points in total, with 74 reference points leading to RNBISub infeasibility. Table 4 shows that Farkas pricing is capable of concluding the infeasibility of 66 out of 74 RNBISub instances using 10 or fewer iterations. The average computation time for these 66 instances is 0.4 s. However, Farkas pricing is incapable of concluding infeasibility within 150 iterations for the remaining 8 instances. The computation time for each of these 8 instances ranges from 1179 to 7084 s, with an average of 4396 s and a standard deviation of 1676 s. For comparison, we apply column generation with the bigM initialisation to 10 reference points leading to RNBISub infeasibility. With a termination condition of 150 column generation iterations, the average computation time for solving each of the 10 reference points is 1213 s, with a standard deviation of 71 seconds. The results suggest that Farkas pricing can potentially be quite time consuming. Thus, if Farkas pricing cannot identify the infeasibility of a RNBISub instance in a small number of iterations, it would be beneficial to change the initialisation method to another approach.
5 Discussion and conclusion
In this paper we propose the use of column generation in the revised normal boundary intersection method to compute a finite representative subset of the nondominated set of a multiobjective linear programme. We introduced a reference point bounding procedure to eliminate the investigation of infeasible reference points. In terms of the quality of a discrete representation computed with the column generation RNBI method we showed that the uniformity level is at least dq, the distance between closest reference points, and therefore the same as that of standard RNBI representative set. The coverage error is bounded by the the distance of CGRNBI points to the nondominated set plus \(\sqrt{p} dq/2\). This feature allows one to choose a value of dq to produce a representative set that suits decision making in the application considered.
To illustrate the method and demonstrate the advantages of using column generation to solve the RNBI subproblems, we apply the method to an MOLP formulation of a radiotherapy treatment design problem, which can be solved by both the standard RNBI and the column generation RNBI method. In agreement with the remark in Lübbecke (2010) that column generation is in general not a competitive technique in solving linear programmes, we observe that computation times using column generation are longer. However, column generation allows us to use variables representing segment intensities, as opposed to bixel intensities which are used in the conventional formulation. As a consequence, the number of variables involved in the model is much greater than for a model formulated with bixel intensities, since the number of possible segment shapes, formed by all possible combinations of opening bixels, is much greater than the number of bixels. On the other hand, the column generation formulation avoids the segmentation step which deteriorates treatment quality. Our results show that plans generated by column generation are nearoptimal and can be delivered with dramatically lower monitor units than the corresponding optimal ones followed by segmentation. This reduced delivery complexity is desirable in practice due to shorter treatment time and lower radiation leakage from the MLCs (hence better delivery accuracy) (Broderick et al. 2009; Carlsson and Forsgren 2014).
Fredriksson and Bokrantz (2013) introduce a concept of nondominance called the “naperture Pareto set”, which is a set of efficient plans given that each plan is formed by only n segments. However, to our knowledge, there is no practical method available to generate the naperture Pareto set. The concept of the naperture Pareto set can be generalised to the ncolumn nondominated set for problems solved by column generation. Further research is required to extend the column generation RNBI method to ensure ncolumn nondominance. Another topic for future research is the extension of the column generation RNBI method to nonlinear multiobjective optimisation problems. This will, e.g., allow us to consider other formulations of the radiotherapy treatment design problem.
Notes
Compliance with ethical standards
Data statement
Due to the sensitive nature of the data, we are not in a position to share data as we do not have permission to do so.
Conflict of interest
The authors declare that they have no conflict of interest.
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