Multi-objective linear programming: Difference between revisions

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'''Multi-objective linear programming''' is a subarea of [[mathematical optimization]]. A multiple objective linear program (MOLP) is a [[linear program]] with more than one linear objective functionsfunction. An MOLP is a special case of a [[vector linear program]]. Multi-objectionobjective linear programming is also a subarea of [[Multi-objective optimization]].

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'''Multi-objective linear programming''' is a subarea of [[mathematical optimization]]. A multiple objective linear program (MOLP) is a [[linear program]] with more than one linear objective functions. An MOLP is a special case of a [[vector linear program]]. Multi-objection linear programming is also a subarea of [[Multi-objective optimization]].

== Problem formulation ==

In mathematical terms, a MOLP can be written as:

:<math>\min_{x}min_x Px \quad\text{s.t.}\quad a \leq Bx \leq b,\; l\ell \leq x \leq u</math>

where <math>B</math> is an <math>(m\times n)</math> matrix, <math>P</math> is a <math>(q\times n)</math> matrix, <math>a</math> is an <math>m</math>-dimensional vector with components in <math>\mathbb{R} \cup \{-\infty\}</math>, <math>b</math> is an <math>m</math>-dimensional vector with components in <math>\mathbb{R} \cup \{+\infty\}</math>, <math>l\ell</math> is an <math>n</math>-dimensional vector with components in <math>\mathbb{R} \cup \{-\infty\}</math>, <math>u</math> is an <math>n</math>-dimensional vector with components in <math>\mathbb{R} \cup \{+\infty\}</math>

== Solution concepts ==

A feasible point <math>x</math> is called ''efficient'' if there is no feasible point <math>y</math> with <math>Px \leq Py</math>, <math>Px \neq Py</math>, where <math>\leq</math> denoteddenotes the component-wise ordering.

Often in the literature, the aim in multiple objective linear programming is to compute the set of all efficient extremal points .....<ref name="EckerKouada1978">{{cite journal|last1=Ecker|first1=J. G.|last2=Kouada|first2=I. A.|title=Finding all efficient extreme points for multiple objective linear programs|journal=Mathematical Programming|volume=14|issue=1|year=1978|pages=249–261|issn=0025-5610|doi=10.1007/BF01588968|s2cid=42726689}}</ref>. There are also algorithms to determine the set of all maximal efficient faces .<ref name="EckerHegner1980">{{cite journal|last1=Ecker|first1=J. G.|last2=Hegner|first2=N. S.|last3=Kouada|first3=I. A.|title=Generating all maximal efficient faces for multiple objective linear programs|journal=Journal of Optimization Theory and Applications|volume=30|issue=3|year=1980|pages=353–381|issn=0022-3239|doi=10.1007/BF00935493|s2cid=120455645}}</ref>. Based on these goals, the set of all efficient (extreme) points can be seen to be the solution of MOLP. This type of solution concept is called ''decision set based''.<ref name="Benson1998">{{cite journal|title=An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem|last1=Benson|first1=Harold P.|journal=Journal of Global Optimization|volume=13|issue=1|year=1998|pages=1–24|issn=092550010925-5001|doi=10.1023/A:1008215702611|s2cid=45440728}}</ref>. It is not compatible with an optimal solution of a linear program but rather parallels the set of all optimal solutions of a linear program (which is more difficult to determine).

Efficient points are frequently called ''efficient solutions''. This term is misleading because a single efficient point can be already obtained by solving one linear program, such as the linear program with the same feasible set and the objective function being the sum of the objectives of MOLP.<ref name="Ehrgott2015">{{cite journalbook|last1=Ehrgott|first1=M.|title=Multicriteria Optimization|journalpublisher=Springer|year=2005|doi=10.1007/3-540-27659-9|isbn=978-3-540-21398-7|citeseerx=10.1.1.360.5223}}</ref>.

More recent references consider ''outcome set based'' solution concepts<ref name="HeydeLöhne2011">{{cite journal|last1=Heyde|first1=Frank|last2=Löhne|first2=Andreas|title=Solution concepts in vector optimization: a fresh look at an old story|journal=Optimization|volume=60|issue=12|year=2011|pages=1421–1440|issn=0233-1934|doi=10.1080/02331931003665108|s2cid=54519405|url=http://webdoc.sub.gwdg.de/ebook/serien/e/reports_Halle-Wittenberg_math/08-19report.pdf}}</ref> and corresponding algorithms.<ref name="DauerSaleh1990">{{cite journal|last1=Dauer|first1=J.P.|last2=Saleh|first2=O.A.|title=Constructing the set of efficient objective values in multiple objective linear programs|journal=European Journal of Operational Research|volume=46|issue=3|year=1990|pages=358–365|issn=037722170377-2217|doi=10.1016/0377-2217(90)90011-Y}}</ref><ref name="Benson1998" /> Assume MOLP is bounded, i.e. there is some <math>y \in \mathbb{R}^q</refmath> such that <math>y \leq Px</math> for all feasible <math>x</math>. A solution of MOLP is defined to be a finite subset <math>\bar S</math> of efficient points that carries a sufficient amount of information in order to describe the ''upper image'' of MOLP:. Denoting by <math>S</math> the feasible set of MOLP by <math>S</math>, the ''upper image'' of MOLP is the set <math>\mathcal{P}:=P[ S] + \mathbb{R}^q_+ := \{ y \in \mathbb{R}^q:\; \exists x \in \mathbb{R}^nS: y \geq Px,\;x\in S \}</math>. A formal definition of a solution <ref name="HeydeLöhne2011">< /ref><ref name="Löhne2011">{{cite book|title=Vector Optimization with Infimum and Supremum|last1=Löhne|first1=Andreas|year=2011|issn=1867-8971|doi=10.1007/978-3-642-18351-5|series=Vector Optimization|isbn=978-3-642-18350-8}}</ref> is as follows:

A finite set <math>\bar S</math> of efficient points is called ''solution'' to MOLP if

<math>\textoperatorname{conv} P[\bar S] + \mathbb{R}^q_+ = \mathcal{P}</math> (''"conv''" denotes the convex hull).

If MOLP is not bounded, a solution consists not only of points but of points and directions <ref name="Löhne2011" /><ref name="LöhneWeißing2017" />

== Solution methods ==

Multiobjective variants of the [[simplex algorithm]] are used to compute decision set based solutions<ref name="EckerKouada1978" /><ref name="EckerHegner1980" /><ref name="ArmandMalivert1991">{{cite journal|last1=Armand|first1=P.|last2=Malivert|first2=C.|title=Determination of the efficient set in multiobjective linear programming|journal=Journal of Optimization Theory and Applications|volume=70|issue=3|year=1991|pages=467–489|issn=0022-3239|doi=10.1007/BF00941298|citeseerx=10.1.1.161.9730|s2cid=18407847}}</ref> and objective set based solutions.<ref name="RudloffUlus2016">{{cite journal|last1=Rudloff|first1=Birgit|last2=Ulus|first2=Firdevs|last3=Vanderbei|first3=Robert|title=A parametric simplex algorithm for linear vector optimization problems|journal=Mathematical Programming|volume=163|issue=1–2|year=2016|pages=213–242|issn=0025-5610|doi=10.1007/s10107-016-1061-z|arxiv=1507.01895|s2cid=13844342}}</ref>

Multiobjective variants of the simplex algorithm are used to compute decision set based solutions<ref name="EckerKouada1978"></ref><ref name="EckerHegner1980"></ref>.

Objective set based solutions can be obtained by [[Benson's algorithm ]].<ref> name="Benson1998"< /ref><ref name="LöhneWeißing2017">{{cite journal|last1=Löhne|first1=Andreas|last2=Weißing|first2=Benjamin|title=The vector linear program solver Bensolve – notes on theoretical background|journal=European Journal of Operational Research|volume=260|issue=3|year=2017|pages=807–813|issn=037722170377-2217|doi=10.1016/j.ejor.2016.02.039|arxiv=15071510.00228v504823|s2cid=17267946}}</ref>.

== Related problem classes ==

Multiobjective linear programming is equivalent to [[Polyhedral combinatorics|polyhedral]] projection.<ref name="LöhneWeißing2016">{{cite journal|last1=Löhne|first1=Andreas|last2=Weißing|first2=Benjamin|title=Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming|journal=Mathematical Methods of Operations Research|volume=84|issue=2|year=2016|pages=411–426|issn=1432-2994|doi=10.1007/s00186-016-0554-0|arxiv=1507.00228|s2cid=26137201}}</ref>.

== References ==

{{Reflist}}

[[Category:MathematicalLinear optimizationprogramming]]