# Time symmetry of resource constrained project scheduling with general temporal constraints and take-give resources

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## Abstract

The paper studies a lacquer production scheduling problem formulated as a resource constrained project scheduling problem with general temporal constraints (i.e., positive and negative time-lags). This real-world scheduling problem requires so-called take-give resources that are needed from the beginning of an activity to the completion of another activity of the production process. Furthermore, we consider sequence dependent changeover times on take-give resources. We formulate this problem by mixed integer linear programming and we suggest a parallel heuristic to solve the problem. This heuristic exploits a time symmetry mapping which allows an easy construction of a schedule in the backward time orientation. In the second part of the paper, it is proven that the time symmetry mapping is bijective and involutive even for the problem with general temporal constraints, changeover times, and take-give resources. The motivation to use this mapping is to improve the performance of the heuristic and to simplify its implementation. Finally, the performance of the heuristic algorithm is evaluated on a set of lacquer production benchmarks requiring take-give resources and on standard benchmarks for the resource constrained project scheduling problem with general temporal constraints where we found new better solutions in 16 and 12 instances out of 90 for UBO500 and UBO1000 respectively.

## Keywords

Scheduling algorithms Project scheduling General temporal constraints Backward scheduling Additional resources## List of symbols

- \(a_{{ ilk}}\)
Binary constant indicating the requirement of take-give resource

*k*for occupation*i*i.e., \(a_{{ ilk}}=1\) iff take-give resource*k*is taken at \(s_i\) and given (released) at \(C_l\)*b*Number of take-give resources

- \(C_i\)
Completion time of activity

*i*- \(C_{{ max}}\)
Schedule makespan

- \(d_{{ ij}}\)
Length of the longest path from node

*i*to node*j**G*Graph

*i*,*j*,*l*,*h*Activity indices

*I*Instance, given by the input parameters of the scheduling problem

*k*Resource index

*m*Number of resources

*n*Number of activities

- \(o_{{ ij}}\)
Changeover time (sequence dependent set-up time) from activity

*i*to activity*j*- \(\tilde{o}_{{ ij}}\)
Changeover time (sequence dependent set-up time) from occupation

*i*to occupation*j*- \(p_i\)
Processing time of activity

*i*- \(\tilde{p}_i\)
Processing time of occupation

*i*i.e., \(\tilde{p}_{i}=C_l - s_i\) iff exists take-give resource*k*taken at \(s_i\) and given at \(C_l\)- \(Q_k\)
Capacity of take-give resource

*k*in number of units- \(r_{{ ik}}\)
Requirement of resource

*k*for activity*i*- \(R_k\)
Capacity of resource

*k*in number of units*S*Schedule, given by variables

*s*,*z*- \(s_i\)
Start time of activity

*i*- \({ UB}\)
Upper bound on schedule makespan

*v*Unit index

- \(x_{{ ij}}\)
Binary variable indicating whether activity

*i*precedes activity*j*on one resource unit- \(\tilde{x}_{{ ij}}\)
Binary variable indicating whether occupation

*i*precedes occupation*j*on one take-give resource unit- \(y_{{ ij}}\)
Binary variable indicating whether activity

*i*and activity*j*must be assigned to different units of common resources i.e. if \(y_{{ ij}}=1\) then \(\forall k \in \left\{ 1,\ldots ,m\right\} , \forall v \in \left\{ 1,\ldots ,R_k\right\} ; z_{{ ivk}} \cdot z_{jvk} \ne 1 \)- \(\tilde{y}_{{ ij}}\)
Binary variable indicating whether occupation

*i*and occupation*j*must be assigned to different units of common take-give resource i.e., if \(\tilde{y}_{{ ij}}=1\) then \(\forall k \in \left\{ 1,\ldots ,b\right\} , \forall v \in \left\{ 1,\ldots ,Q_k\right\} ; \tilde{z}_{{ ivk}} \cdot \tilde{z}_{jvk} \ne 1 \)- \(z_{{ ivk}}\)
assignment binary variable i.e., \(z_{{ ivk}}=1\) iff activity

*i*is assigned to unit*v*of resource*k*- \(\tilde{z}_{{ ivk}}\)
Take-give assignment binary variable i.e., \(\tilde{z}_{{ ivk}}=1\) iff occupation

*i*is assigned to unit*v*of take-give resource*k*- \(\delta _{{ ij}}\)
Length of edge \(e_{{ ij}}\)

- \(\mathcal {E}\)
Set of edges \(e_{{ ij}}\)

- \(\mathcal {M}\)
Set of resource conflicts between two activities i.e., unordered couple \(\left\{ i,j\right\} \in \mathcal {M}\) iff \(\exists k \in \left\{ 1,\ldots ,m \right\} ; r_{{ ik}} \cdot r_{jk} \ge 1\) and \(R_k < \infty \) and \((~d_{{ ij}} < (p_i + o_{{ ij}})\) and \(~d_{ji} < (p_j + o_{ji}))\)

- \(\mathcal {Q}\)
Set of take-give resources

- \(\mathcal {R}\)
Set of resources

- \(\mathcal {S}\)
Set of feasible schedules

- \(\mathcal {V}\)
Set of activities

## Notes

### Acknowledgments

The authors would like to thank Mirko Navara from the Czech Technical University and anonymous reviewers for their valuable comments and inspirational suggestions. This work was supported by the Grant Agency of the Czech Republic under the Project GACR P103-16-23509S.

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