ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Multi-objective linear programming</title>

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Multi-objective linear programming

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">Multi-objective linear programming is a subarea of <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>. A multiple objective linear program (MOLP) is a <a href="Linear_program" class="mw-redirect" title="Linear program">linear program</a> with more than one objective function. An MOLP is a special case of a vector linear program. Multi-objective linear programming is also a subarea of <a href="Multi-objective_optimization" title="Multi-objective optimization">Multi-objective optimization</a>.

<div class="mw-heading mw-heading2"><h2 id="Problem_formulation">Problem formulation</h2></div>

In mathematical terms, a MOLP can be written as:

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \min _{x}Px\quad {\text{s.t.}}\quad a\leq Bx\leq b,\;\ell \leq x\leq u}</annotation>

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</math><img src="./fc2f7fbab835f70a310dfe0454876ee2fa675396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.12ex; height:3.676ex;" alt="{\displaystyle \min _{x}Px\quad {\text{s.t.}}\quad a\leq Bx\leq b,\;\ell \leq x\leq u}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m\times n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m\times n)}</annotation>

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</math><img src="./aba857b5d17e977455273ca9af1baf4da35de324.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.085ex; height:2.843ex;" alt="{\displaystyle (m\times n)}" loading="lazy"> matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q\times n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (q\times n)}</annotation>

</semantics>

</math><img src="./31fe653108d861f6f3150a8efca77bdf2a930e25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.114ex; height:2.843ex;" alt="{\displaystyle (q\times n)}" loading="lazy"> matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-dimensional vector with components in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{-\infty \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{-\infty \}}</annotation>

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</math><img src="./d803f13229b881b5f52a24255a3a74f0368af546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{-\infty \}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-dimensional vector with components in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{+\infty \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{+\infty \}}</annotation>

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</math><img src="./3f5ee15b830c1d223d89dd5c92b99f213a8d88ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{+\infty \}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

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<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

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</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional vector with components in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{-\infty \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{-\infty \}}</annotation>

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</math><img src="./d803f13229b881b5f52a24255a3a74f0368af546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{-\infty \}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional vector with components in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \cup \{+\infty \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \cup \{+\infty \}}</annotation>

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</math><img src="./3f5ee15b830c1d223d89dd5c92b99f213a8d88ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \cup \{+\infty \}}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Solution_concepts">Solution concepts</h2></div>

A feasible point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is called efficient if there is no feasible point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Px\leq Py}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Px\leq Py}</annotation>

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</math><img src="./7173be5655b991e8fada986d0916bcdb585ffe5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.074ex; height:2.509ex;" alt="{\displaystyle Px\leq Py}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Px\neq Py}">

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<annotation encoding="application/x-tex">{\displaystyle Px\neq Py}</annotation>

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</math><img src="./ac1bdbae6b180e3d558628ffb5d236e74d868010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.074ex; height:2.676ex;" alt="{\displaystyle Px\neq Py}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq }</annotation>

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</math><img src="./440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> denotes 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.....<a href="#cite_note-EckerKouada1978-1">[1]</a> There are also algorithms to determine the set of all maximal efficient faces.<a href="#cite_note-EckerHegner1980-2">[2]</a> 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.<a href="#cite_note-Benson1998-3">[3]</a> 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.<a href="#cite_note-Ehrgott2015-4">[4]</a>

More recent references consider outcome set based solution concepts<a href="#cite_note-HeydeLöhne2011-5">[5]</a> and corresponding algorithms.<a href="#cite_note-DauerSaleh1990-6">[6]</a><a href="#cite_note-Benson1998-3">[3]</a> Assume MOLP is bounded, i.e. there is some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {R} ^{q}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {R} ^{q}}</annotation>

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</math><img src="./1e5dc13ea04ba709ab44a2b4e3787960c0c1d641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.663ex; height:2.676ex;" alt="{\displaystyle y\in \mathbb {R} ^{q}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq Px}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\leq Px}</annotation>

</semantics>

</math><img src="./efdffeb917197304114b600c7fb67c0f9c133dd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.329ex; height:2.509ex;" alt="{\displaystyle y\leq Px}" loading="lazy"> for all feasible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. A solution of MOLP is defined to be a finite subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {S}}}</annotation>

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</math><img src="./8477edc5727710564fab7b520bd09d19b1fdc95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.573ex; height:2.676ex;" alt="{\displaystyle {\bar {S}}}" loading="lazy"> of efficient points that carries a sufficient amount of information in order to describe the upper image of MOLP. Denoting by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> the feasible set of MOLP, the upper image of MOLP is the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}:=P[S]+\mathbb {R} _{+}^{q}:=\{y\in \mathbb {R} ^{q}:\;\exists x\in S:y\geq Px\}}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}:=P[S]+\mathbb {R} _{+}^{q}:=\{y\in \mathbb {R} ^{q}:\;\exists x\in S:y\geq Px\}}</annotation>

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</math><img src="./46920c08ee1b648e0025ddc4554b88fe0496d583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.56ex; height:3.176ex;" alt="{\displaystyle {\mathcal {P}}:=P[S]+\mathbb {R} _{+}^{q}:=\{y\in \mathbb {R} ^{q}:\;\exists x\in S:y\geq Px\}}" loading="lazy">. A formal definition of a solution <a href="#cite_note-HeydeLöhne2011-5">[5]</a><a href="#cite_note-Löhne2011-7">[7]</a> is as follows:

A finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {S}}}</annotation>

</semantics>

</math><img src="./8477edc5727710564fab7b520bd09d19b1fdc95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.573ex; height:2.676ex;" alt="{\displaystyle {\bar {S}}}" loading="lazy"> of efficient points is called solution to MOLP if

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {conv} P[{\bar {S}}]+\mathbb {R} _{+}^{q}={\mathcal {P}}}</annotation>

</semantics>

</math><img src="./0cd4f2c1d4eeb20af1d31d37b6388192bf267746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.545ex; height:3.343ex;" alt="{\displaystyle \operatorname {conv} P[{\bar {S}}]+\mathbb {R} _{+}^{q}={\mathcal {P}}}" loading="lazy"> ("conv" denotes the convex hull).

If MOLP is not bounded, a solution consists not only of points but of points and directions <a href="#cite_note-Löhne2011-7">[7]</a><a href="#cite_note-LöhneWeißing2017-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Solution_methods">Solution methods</h2></div>

Multiobjective variants of the <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a> are used to compute decision set based solutions<a href="#cite_note-EckerKouada1978-1">[1]</a><a href="#cite_note-EckerHegner1980-2">[2]</a><a href="#cite_note-ArmandMalivert1991-9">[9]</a> and objective set based solutions.<a href="#cite_note-RudloffUlus2016-10">[10]</a>

Objective set based solutions can be obtained by <a href="Benson's_algorithm" title="Benson's algorithm">Benson's algorithm</a>.<a href="#cite_note-Benson1998-3">[3]</a><a href="#cite_note-LöhneWeißing2017-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Related_problem_classes">Related problem classes</h2></div>

Multiobjective linear programming is equivalent to <a href="Polyhedral_combinatorics" title="Polyhedral combinatorics">polyhedral</a> projection.<a href="#cite_note-LöhneWeißing2016-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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