Journal of Scheduling

, Volume 21, Issue 5, pp 545–563 | Cite as

A new algorithm based on evolutionary computation for hierarchically coupled constraint optimization: methodology and application to assembly job-shop scheduling

  • Pan Zou
  • Manik Rajora
  • Steven Y. LiangEmail author


Hierarchically coupled constraint optimization (HCCO) problems are omnipresent, both in theoretical problems and in real-life scenarios; however, there is no clear definition to identify these problems. Numerous techniques have been developed for some typical HCCO problems, such as assembly job-shop scheduling problems (AJSSPs); however, these techniques are not universally applicable to all HCCO problems. In this paper, an abstract definition and common principles amongst different HCCO problems are first established. Next, based on the definitions and principles, a new optimization algorithm based on evolutionary computation is developed for HCCO. The new optimization algorithm has three key new features: a new initial solution generator, a level barrier-based crossover operator, and a level barrier-based mutation operator. In the initial solution generator, a partial solution is created in the first step that satisfies the lowest level hierarchically coupled constraint (HCC) and each consecutive step afterwards adds on to the partial solution to satisfy the next higher level of HCC. In the level barrier-based operators, the operations are only performed between genes satisfying the same level of HCCs to ensure feasibility of the new solutions. The developed optimization algorithm is used to solve a variety of AJSSPs and the results obtained using the proposed algorithm are compared to other methods used to solve AJSSPs.


Evolutionary computation Hierarchically coupled constraint optimization Multi-chromosome Assembly job-shop scheduling 

List of symbols


\(n{\hbox {th}}\) independent variable


The frequency of occurrence of the variable \(x(a_i)\)

\({{\varvec{b}}}_{{{\varvec{m}}}_{{\varvec{1}}}}^{\left( \mathbf{1} \right) }\)

\(m_1\)-dependent variable belonging to the \(1{\mathrm{st}}\) level


Constraint on the independent variable \(a_1\)


Constraint on the dependent \(b_1\)


Location of variable \(a_i\) in the chromosome

\({{\varvec{l}}}_{{{\varvec{b}}}_\mathbf{1}^{\left( \mathbf{1} \right) }}\)

Location of variable \(b_1^{\left( 1 \right) }\) in the chromosome

\({{\varvec{G}}}_{{{\varvec{j}}}_\mathbf{1}}^{\left( \mathbf{0} \right) }\)

\(j_1 \; 0{\mathrm{th}}\)-level gene

\({{\varvec{w}}}_{{{\varvec{j}}}_\mathbf{1}, {{\varvec{i}}}}^{\left( \mathbf{0} \right) }\)

Weight values of the independent variable \(a_i\) for gene \(G_{j_1 }^{\left( 0 \right) }\)

\({{\varvec{w}}}_{{{\varvec{j}}}_\mathbf{2}, {{\varvec{j}}}_\mathbf{1}}^{\left( \mathbf{1} \right) }\)

Weight values of the \(1{\mathrm{st}}\)-level dependent variable \(b_{j_1 }^{\left( 1 \right) }\) for gene \(G_{j_1 }^{\left( 1 \right) }\)


Crossover weight values

\({{\varvec{w}}}_{{\varvec{x}}} \left( {{{\varvec{r}}}_\mathbf{0} } \right) \)

Crossover weight value for the \(r_0{\hbox {th}}\) level


Mutation-level weight values

\({{\varvec{w}}}_{{\varvec{m}}} \left( {{{\varvec{r}}}_\mathbf{0}} \right) \)

Mutation weight value for the \(r_0{\hbox {th}}\) level


Random integer between 1 and \(m_{r_0}\)

\({{\varvec{CM}}}^{\left( \mathbf{0} \right) }\)

Mutation candidates for the \(0{\mathrm{th}}\) level


Mutation candidates for all levels

\({{\varvec{\hbox {CG}}}}_{{{\varvec{b}}}_{{{\varvec{m}}}_{\left( {{{\varvec{r}}}_\mathbf{0} +\mathbf{1}} \right) } }^{\left( {{{\varvec{r}}}_\mathbf{0} +\mathbf{1}} \right) } }^{\left( {{{\varvec{r}}}_\mathbf{0}} \right) }\)

A vector containing gene groups belonging to the \(r_{o}\) level that constrains the \(b_{m_{\left( {r_0 +1} \right) } }^{\left( {r_0 +1} \right) }\) dependent variable and has the length c


Mutation probability (between 0 and 1)


Random integer between 1 and \(m_{\left( {r_0 +1} \right) }\)


Random integer between 1 and number of genes in c


Random integer between 2 and the number of genes in \({\hbox {CG}}_{b_{m_{\left( P \right) } }^{\left( {r_0 +1} \right) } }^{\left( {r_0 } \right) } \left( Q \right) -R\)


Number of genes in \({\hbox {CG}}_{b_{m_{\left( P \right) } }^{\left( {r_0 -1} \right) } }^{\left( {r_0 } \right) } \left( Q \right) -R\)


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ComputingGeorgia Institute of TechnologyAtlantaUSA
  2. 2.George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

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