A Hybrid CLP/MP Approach to Modeling and Solving Resource-Constrained Scheduling Problems with Logic Constraints

Abstract

Constrained scheduling problems are common in everyday life and especially in: distribution, manufacturing, project management, logistics, supply chain management, software engineering, computer networks etc. A large number of integer and binary decision variables representing the allocation of different constrained resources to activities/jobs and constraints on these decision variables are typical elements of the resource-constrained scheduling problems (RCSPs) modeling. Therefore, the models of RCSPs are more demanding, particularly when methods of operations research (OR) are used. By contrast, most resource-constrained scheduling problems can be easily modeled as instances of the constraint satisfaction problems (CSPs) and solved using constraint logic programming (CLP) or others methods. Moreover, CLP-based environments enable easy modeling of various types of constraints including logic constraints.

In the CLP-based environment the problem definition is separated from the algorithms and methods used to solve the problem. Therefore, a hybrid approach to resource-constrained scheduling problems that combines an OR-based approach for problem solving and a CLP-based approach for problem modeling is proposed. To evaluate the efficiency and flexibility of this approach, illustrative examples of resource-constrained scheduling problems with logic constraints are implemented using hybrid CLP/MP approach.

Appendices

Appendix A Sets of Facts for Illustrative Example

Appendix B Illustrative Example-Formal Model

Table B1. Indices, parameters and constraints for mathematical model of RCSP

$$ {\text{Min C}}_{ \hbox{max} } $$

(1)

$$ {\text{G}}_{{{\text{k}},{\text{n}}}} \le {\text{C}}_{\hbox{max} } \forall {\text{k}} = 1..{\text{K}},{\text{n}} = 1..{\text{N}} $$

(2)

$$ \sum\limits_{{{\text{e}} = 1}}^{\text{E}} {\sum\limits_{{{\text{p}} = 1}}^{\text{P}} {{\text{e}}1_{{{\text{k}},{\text{n}}}} \cdot {\text{cu}}_{{{\text{k}},{\text{n}},{\text{e}}}} \cdot {\text{X}}_{{{\text{k}},{\text{n}},{\text{e}},{\text{p}}}} } } = {\text{f}}_{{{\text{k}},{\text{n}}}} \cdot {\text{r}}_{\text{k}} \forall {\text{k}} = 1..{\text{K}},{\text{n}} = 1..{\text{N}} $$

(3)

$$ \sum\limits_{{{\text{k}} = 1}}^{\text{K}} {\sum\limits_{{{\text{e}} = 1}}^{\text{E}} {{\text{X}}_{{{\text{k}},{\text{n}},{\text{e}},{\text{p}}}} } } \le 1\forall {\text{n}} = 1..{\text{N}},{\text{p}} = 1..{\text{P}} $$

(4)

$$ \sum\limits_{{{\text{k}} = 1}}^{\text{K}} {\sum\limits_{{{\text{n}} = 1}}^{\text{N}} {{\text{c}}_{{{\text{k}},{\text{n}},{\text{e}}}} \cdot {\text{X}}_{{{\text{k}},{\text{n}},{\text{e}},{\text{p}}}} } } \le {\text{v}}_{\text{e}} \forall {\text{e}} = 1..{\text{E}},{\text{p}} = 1..{\text{P}} $$

(5)

$$ {\text{X}}_{{{\text{k}},{\text{n}},{\text{e}},{\text{p}} - 1}} - {\text{X}}_{{{\text{k,n}},{\text{e,p}}}} \le {\text{Z}}_{{{\text{k,n,e,p}} - 1}} \forall {\text{k}} = 1..{\text{K}},{\text{n}} = 1..{\text{N}},{\text{e}} = 1..{\text{E}},{\text{p}} = 2..{\text{P}} $$

(6)

$$ \sum\limits_{{{\text{p}} = 1}}^{\text{P}} {{\text{Z}}_{\text{k,n,e,p}} } = 1\forall {\text{k}} = 1..{\text{K}},{\text{n}} = 1..{\text{N}},{\text{s}} = {\text{e}}..{\text{E}} $$

(7)

$$ {\text{G}}_{\text{k,n}} \ge {\text{h}}_{\text{p}} \cdot {\text{X}}_{\text{k,n,e,p}} \forall {\text{k}} = 1..{\text{K}},{\text{n}} = 1..{\text{N}},{\text{e}} = 1..{\text{E}},{\text{p}} = 1..{\text{P}} $$

(8)

$$ {\text{G}}_{\text{k,n2}} - {\text{f}}_{\text{k,n2}} \ge {\text{G}}_{\text{k,n1}} \forall {\text{k}} = 1..{\text{K}},{\text{n}}1,{\text{n}}2 = 1..{\text{N:lo}}_{\text{k,n1,n2}} = 1 $$

(9)

$$ {\text{G}}_{\text{k,n}} \in {\text{C}}\forall {\text{k}} = 1..{\text{K,n}} = 1..{\text{N}} $$

(10)

$$ {\text{X}}_{\text{k,n,e,p}} \in \{ 0,1\} \forall {\text{k}} = 1..{\text{K,n}} = 1..{\text{N}},{\text{e}} = 1..{\text{E}},{\text{p}} = 1..{\text{P}} $$

(11)

$$ {\text{Z}}_{\text{k,n,e,p}} \in \{ 0,1\} \forall {\text{k}} = 1..{\text{K}},{\text{n}} = 1..{\text{N}},{\text{e}} = 1..{\text{E}},{\text{p}} = 1..{\text{P}} $$

(12)

$$ {\text{Exclusion}}\_{\text{N(n1,n2) }}\exists {\text{n}}1,{\text{n}}2 = 1..{\text{N:n}}1 \ne {\text{n}}2 $$

(13)

$$ {\text{Exclusion}}\_{\text{R(e1,e2) }}\exists {\text{e}}1,{\text{e}}2 = 1..{\text{E:e}}1 \ne {\text{e}}2 $$

(14)

$$ {\text{Exclusion}}\_{\text{NR(n1,e2) }}\exists {\text{n}} = 1..{\text{N}},{\text{e}} = 1..{\text{E}} $$

(15)