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A Constraint-Based Framework for Scheduling Problems

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Intelligent Information and Database Systems (ACIIDS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10751))

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Abstract

Scheduling and resource allocation problems are widespread in many areas of today’s technology and management. Their different forms and structures appear in production, logistics, software engineering, computer networks, etc. In practice, however, classical scheduling problems with fixed structures and only standard constraints (precedence, disjoint etc.) are rare. Practical scheduling problems include also logical and non-linear constraints and use non-standard criteria of schedule evaluations. In many cases, decision makers are interested in the feasibility and/or optimality of a given schedule for specified conditions formulated as questions, for example, Is it possible…?, What is the minimum/maximum…?, What if..? etc. Thus there is a need to develop a programming framework that will facilitate the modeling and solving a variety of diverse scheduling problems. This paper proposes such a constraint-based framework for modeling and solving scheduling problems. It was built with the CLP (Constraint Logic Programming) environment and supported with MP (Mathematical Programming).

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Author information

Authors and Affiliations

  1. Institute of Management and Control Systems, Kielce University of Technology, Al. 1000-lecia PP 7, 25-314, Kielce, Poland

    Jarosław Wikarek, Paweł Sitek & Tadeusz Stefański

Authors
  1. Jarosław Wikarek
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  2. Paweł Sitek
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  3. Tadeusz Stefański
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Corresponding author

Correspondence to Paweł Sitek .

Editor information

Editors and Affiliations

  1. Wrocław University of Science and Technology, Wrocław, Poland

    Ngoc Thanh Nguyen

  2. Quang Binh University, Dong Hoi City, Vietnam

    Duong Hung Hoang

  3. National University of Kaohsiung, Kaohsiung, Taiwan

    Tzung-Pei Hong

  4. Rutgers University, Piscataway, New Jersey, USA

    Hoang Pham

  5. Wrocław University of Science and Technology, Wrocław, Poland

    Bogdan Trawiński

Appendix A Models and Facts

Appendix A Models and Facts

(See Tables 5, 6, 7 and 8).

Table 7. Description of the constraints for both models
Full size table
Table 8. Description of the facts
Full size table
$$ {\text{Wp}} = \sum\limits_{{{\text{b}} = 1}}^{\text{LB}} {\sum\limits_{{{\text{a}} = 1}}^{\text{LA}} {\sum\limits_{{{\text{c}} = 1}}^{\text{LC}} {\sum\limits_{{{\text{d}} = 1}}^{\text{LD}} {{\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} \cdot } {\text{ro}}_{\text{c}} } } } $$
(1)
$$ {\text{Z}}_{{{\text{b}},{\text{a}}}} \le {\text{Cmax}} \forall {\text{b}} = 1..{\text{LB}},{\text{a}} = 1..{\text{LA}} $$
(2)
$$ \sum\limits_{{{\text{c}} = 1}}^{\text{LC}} {\sum\limits_{{{\text{d}} = 1}}^{\text{LD}} {{\text{bo}}_{{{\text{b}},{\text{a}},{\text{c}}}} \cdot {\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} } } = {\text{ex}}_{{{\text{b}},{\text{a}}}} \cdot {\text{za}}_{\text{b}} \forall {\text{b}} = 1..{\text{LB}},{\text{a}} = 1..{\text{LA}}:{\text{ex}}_{{{\text{b}},{\text{a}}}} > 0 $$
(3)
$$ \sum\limits_{{{\text{b}} = 1}}^{\text{LB}} {\sum\limits_{{{\text{c}} = 1}}^{\text{LC}} {{\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} } } \le 1\forall {\text{a}} = 1..{\text{LA}},{\text{d}} = 1..{\text{LD}} $$
(4)
$$ \sum\limits_{a = 1}^{\text{LA}} {\sum\limits_{{{\text{b}} = 1}}^{\text{LB}} {{\text{bo}}_{{{\text{b}},{\text{a}},{\text{c}}}} \cdot {\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} } } \le {\text{fo}}_{\text{c}} \forall {\text{c}} = 1..{\text{LC}},{\text{d}} = 1..{\text{LD}} $$
(5)
$$ {\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}} - 1}} - {\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} \le {\text{Y}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}} - 1}} \forall {\text{a}} = 1..{\text{LA}},{\text{b}} = 1..{\text{LB}},{\text{c}} = 1..{\text{LC}},{\text{d}} = 2..{\text{LD}} $$
(6)
$$ \begin{array}{*{20}c} {\sum\limits_{{{\text{d}} = 1}}^{\text{LD}} {\sum\limits_{{{\text{c}} = 1}}^{\text{LC}} {{\text{Y}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} } } = 1\forall {\text{a}} = 1..{\text{LA}},{\text{b}} = 1..{\text{LB}}:{\text{ti}}_{\text{ba}} > 0} \\ {{\text{Y}}_{{{\text{b}},{\text{a}},{\text{c}},1}} = 0\forall {\text{a}} = 1..{\text{LA}},{\text{b}} = 1..{\text{LB}},{\text{C}} = 1..{\text{LC}},{\text{Y}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{LD}}}} = 0\forall {\text{a}} = 1..{\text{LA}},{\text{b}} = 1..{\text{LB}},{\text{C}} = 1..{\text{LC}}} \\ \end{array} $$
(7)
$$ {\text{Z}}_{{{\text{b}},{\text{a}}}} = \sum\limits_{{{\text{c}} = 1}}^{\text{LC}} {\sum\limits_{{{\text{d}} = 1}}^{\text{LD}} {({\text{pr}}_{\text{d}} \cdot {\text{Y}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} )} } \forall {\text{a}} = 1..{\text{LA}},{\text{b}} = 1..{\text{LB}}:{\text{ex}}_{{{\text{b}},{\text{a}}}} > 0 $$
(8)
$$ {\text{Z}}_{{{\text{b}},{\text{a}}2}} - {\text{za}}_{\text{b}} \cdot {\text{ex}}_{{{\text{b}},{\text{a}}2}} \ge {\text{Z}}_{{{\text{b}},{\text{a}}1}} \forall {\text{b}} = 1..{\text{LB}},{\text{a}}1,{\text{a}}2 = 1..{\text{LA}}:{\text{go}}_{{{\text{b}},{\text{a}}1,{\text{a}}2}} = 1 $$
(9)
$$ {\text{X}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} \in \{ 0,1\} \forall {\text{b}} = 1..{\text{LB}},{\text{a}} = 1..{\text{LA}},{\text{c}} = 1..{\text{LC}},{\text{d}} = 1..{\text{LD}};{\text{Y}}_{{{\text{b}},{\text{a}},{\text{c}},{\text{d}}}} \in \{ 0,1\} \forall {\text{b}} = 1..{\text{LB}},{\text{a}} = 1..{\text{LA}},{\text{c}} = 1..{\text{LC}},{\text{d}} = 1..{\text{LD}} $$
(10)
$$ {\text{Wp}} = \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {{\text{X}}_{{{\text{f}},{\text{d}}}} \cdot {\text{ro}}_{\text{f}} } $$
(1T)
$$ {\text{Z}}_{\text{f}} \le {\text{Cmax}} \forall {\text{f}} = 1..{\text{LF}} $$
(2T)
$$ \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {\sum\limits_{{{\text{d}} = 1}}^{\text{LD}} {{\text{ist}}3_{{{\text{f}},{\text{b}}}} \cdot {\text{ist}}2_{{{\text{f}},{\text{a}}}} \cdot {\text{X}}_{{{\text{f}},{\text{d}}}} } } = {\text{ex}}_{{{\text{b}},{\text{a}}}} \cdot {\text{za}}_{\text{b}} \forall {\text{b}} = 1..{\text{LB}},{\text{a}} = 1..{\text{LA}}:{\text{ex}}_{{{\text{b}},{\text{a}}}} > 0 $$
(3T)
$$ \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {{\text{ist}}2_{{{\text{f}},{\text{a}}}} \cdot {\text{X}}_{{{\text{f}},{\text{d}}}} } \le 1\forall {\text{a}} = 1..{\text{LA}},{\text{d}} = 1..{\text{LD}} $$
(4T)
$$ \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {{\text{ist}}1_{{{\text{f}},{\text{c}}}} \cdot {\text{bo}}_{{{\text{f}},{\text{c}}}} \cdot {\text{X}}_{{{\text{f}},{\text{d}}}} } \le {\text{fo}}_{\text{c}} \forall {\text{c}} = 1..{\text{LC}},{\text{d}} = 1..{\text{LD}} $$
(5T)
$$ {\text{X}}_{{{\text{f}},{\text{d}} - 1}} - {\text{X}}_{{{\text{f}},{\text{d}}}} \le {\text{Y}}_{{{\text{f}},{\text{d}} - 1}} \forall {\text{f}} = 1..{\text{F}},{\text{d}} = 2..{\text{LD}} $$
(6T)
$$ \sum\limits_{{{\text{d}} = 1}}^{\text{LD}} {\sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {{\text{ist}}2_{{{\text{f}},{\text{a}}}} \cdot {\text{ist}}3_{{{\text{f}},{\text{b}}}} \cdot {\text{Y}}_{{{\text{f}},{\text{d}}}} } } \le 1\forall {\text{a}} = 1..{\text{FA}},{\text{b}} = 1..{\text{FB}}:{\text{ex}}_{{{\text{b}},{\text{a}}}} > 0{\text{Y}}_{{{\text{f}},1}} = 0\forall {\text{f}} = 1..{\text{LFY}}_{{{\text{f}}.,{\text{LD}}}} = 0 $$
(7T)
$$ {\text{Z}}_{\text{f}} = \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {{\text{pr}}_{\text{d}} \cdot {\text{X}}_{{{\text{f}},{\text{d}}}} } \forall {\text{f}} = 1..{\text{LF}} $$
(8T)
$$ \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {({\text{ist}}2_{{{\text{f}},{\text{a}}2}} \cdot {\text{ist}}3_{{{\text{f}},{\text{b}}}} \cdot {\text{Z}}_{\text{f}} )} - {\text{za}}_{\text{b}} \cdot {\text{ex}}_{{{\text{b}},{\text{a}}2}} \ge \sum\limits_{{{\text{f}} = 1}}^{\text{LF}} {({\text{ist}}2_{{{\text{f}},{\text{a}}1}} \cdot {\text{ist}}3_{{{\text{f}},{\text{b}}}} \cdot {\text{Z}}_{\text{f}} )} \forall {\text{b}} = 1..{\text{LB}},{\text{a}}1,{\text{a}}2 = 1..{\text{LA}}:{\text{go}}_{{{\text{b}},{\text{a}}1,{\text{a}}2}} = 1 $$
(9T)
$$ {\text{X}}_{{{\text{f}},{\text{d}}}} \in \{ 0,1\} \forall {\text{f}} = 1..{\text{LF}},{\text{d}} = 1..{\text{LD}};{\text{Y}}_{{{\text{f}},{\text{d}}}} \in \{ 0,1\} \forall {\text{f}} = 1..{\text{LF}},{\text{d}} = 1..{\text{LD}} $$
(10T)

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Wikarek, J., Sitek, P., Stefański, T. (2018). A Constraint-Based Framework for Scheduling Problems. In: Nguyen, N., Hoang, D., Hong, TP., Pham, H., Trawiński, B. (eds) Intelligent Information and Database Systems. ACIIDS 2018. Lecture Notes in Computer Science(), vol 10751. Springer, Cham. https://doi.org/10.1007/978-3-319-75417-8_40

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  • DOI: https://doi.org/10.1007/978-3-319-75417-8_40

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