An MP/CP-Based Hybrid Approach to Optimization of the Resource-Constrained Scheduling Problems

Abstract

Constrained scheduling problems are quite common in project management, manufacturing, distribution, transportation, logistics, supply chain management, software engineering, and computer networks etc. The need to use integer and binary decision variables representing the allocation of different resources to many activities and numerous specific, universal and additional constraints on these decision variables are typical components of the resource-constrained scheduling problems (RCSPs) modeling. It is often necessary to model additional resources and constraints. For these reasons, models are becoming computationally demanding. This is particularly noticeable when methods of operations research (mathematical programing (MP), network programing, and dynamic programming) are used. On the other hand, most RCPSs can be easily modeled as instances of the constraint satisfaction problems (CSPs) and solved using constraint programming (CP) methods. In contrast to the MP-based environment, the CP-based environment methods deal well with binary constraints but worse in optimization. Therefore, the hybrid approach based on integration mathematical programming and constraint logic programming to optimization resource-constrained scheduling problems has been proposed. To evaluate the applicability and computational efficiency of the proposed approach and its implementation programming framework, the illustrative examples of optimization resource-constrained scheduling problems are implemented separately for mathematical programming and hybrid method.

Appendices

Appendix A. Sets of Facts for Illustrative Example

Appendix B. Illustrative Example-Formal Model

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

(f1)

$$ {\text{G}}_{{{\text{j}},{\text{f}}}} \le {\text{C}}_{ \hbox{max} } \forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}} $$

(o2)

$$ \sum\limits_{{{\text{a}} = 1}}^{\text{A}} {\sum\limits_{{{\text{t}} = 1}}^{\text{T}} {{\text{d}}1_{{{\text{j}},{\text{f}}}} \cdot {\text{bu}}_{{{\text{j}},{\text{f}},{\text{a}}}} \cdot {\text{X}}_{{{\text{j,f,a}},{\text{t}}}} } } = {\text{d}}_{{{\text{j}},{\text{f}}}} \cdot {\text{o}}_{\text{j}} \forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}} $$

(o3)

$$ \sum\limits_{{{\text{j}} = 1}}^{\text{J}} {\sum\limits_{{{\text{a}} = 1}}^{\text{A}} {{\text{X}}_{{{\text{j,f,a}},{\text{t}}}} } } \le 1\forall {\text{f}} = 1..{\text{F}},{\text{t}} = 1..{\text{T}} $$

(o4)

$$ \sum\limits_{{{\text{j}} = 1}}^{\text{J}} {\sum\limits_{{{\text{f}} = 1}}^{\text{F}} {{\text{b}}_{\text{j,f,a}} \cdot {\text{X}}_{{{\text{j,f,a}},{\text{t}}}} } } \le {\text{c}}_{\text{a}} \forall {\text{a}} = 1..{\text{A}},{\text{t}} = 1..{\text{T}} $$

(o5)

$$ {\text{X}}_{{{\text{j}},{\text{f}},{\text{a}},{\text{t}} - 1}} - {\text{X}}_{{{\text{j,f,a}},{\text{t}}}} \le {\text{Z}}_{{{\text{j,f,a}},{\text{t}} - 1}} \forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}},{\text{a}} = 1..{\text{A}},{\text{t}} = 2..{\text{T}} $$

(o6)

$$ \sum\limits_{{{\text{t}} = 1}}^{\text{T}} {{\text{Z}}_{{{\text{j,f,a}},{\text{t}}}} } = 1\forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}},{\text{a}} = 1..{\text{A}} $$

(o7)

$$ {\text{G}}_{{{\text{j}},{\text{f}}}} \ge {\text{e}}_{\text{t}} \cdot {\text{X}}_{{{\text{j,f,a}},{\text{t}}}} \forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}},{\text{a}} = 1..{\text{A}},{\text{t}} = 1..{\text{T}} $$

(o8)

$$ {\text{G}}_{{{\text{j}},{\text{f}}2}} - {\text{b}}_{{{\text{j}},{\text{f}}2}} \ge {\text{G}}_{{{\text{j}},{\text{f}}1}} \forall {\text{j}} = 1..{\text{J}},{\text{f}}1,{\text{f}}2 = 1..{\text{F}}:{\text{ko}}_{{{\text{j}},{\text{f}}1,{\text{f}}2}} = 1 $$

(o9)

$$ {\text{G}}_{{{\text{j}},{\text{f}}}} \in {\text{C}}\forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}} $$

(o10)

$$ {\text{X}}_{{{\text{j,f,a}},{\text{t}}}} \in \{ 0,1\} \forall {\text{u}} = {\text{j}}..{\text{J}},{\text{f}} = 1..{\text{F}},{\text{a}} = 1..{\text{A}},{\text{t}} = 1..{\text{T}} $$

(o11)

$$ {\text{Z}}_{{{\text{j}},{\text{f}},{\text{a}},{\text{t}}}} \in \{ 0,1\} \forall {\text{j}} = 1..{\text{J}},{\text{f}} = 1..{\text{F}},{\text{a}} = 1..{\text{A}},{\text{t}} = 1..{\text{T}} $$

(o12)

$$ {\text{Exclusion}}\_{\text{M(f1,f2) }}\exists {\text{f}}1,{\text{f}}2 = 1..{\text{F}}:{\text{f}}1 \ne {\text{f}}2 $$

(o13)

$$ {\text{Exclusion}}\_{\text{R(a1,a2) }}\exists {\text{a}}1,{\text{a}}2 = 1..{\text{A}}:{\text{a}}1 \ne {\text{a}}2 $$

(o14)

$$ { \hbox{min} }\,\sum\limits_{{{\text{j}} = 1}}^{\text{J}} {\sum\limits_{{{\text{f}} = 1}}^{\text{F}} {\sum\limits_{{{\text{a}} = 1}}^{\text{A}} {\sum\limits_{{{\text{t}} = 1}}^{\text{T}} {({\text{X}}_{{{\text{j,f,a}},{\text{t}}}} \cdot {\text{r}}_{\text{a}} )} } } } $$

(f2)

$$ {\text{min U}}_{\text{a}} = \sum\limits_{{{\text{j}} = 1}}^{\text{J}} {\sum\limits_{{{\text{f}} = 1}}^{\text{F}} {\sum\limits_{{{\text{t}} = 1}}^{\text{T}} {({\text{X}}_{{{\text{j,f,a}},{\text{t}}}} )} \forall {\text{a}} = 1..{\text{A}}} } $$

(f3)