# A linear time–cost tradeoff problem with multiple milestones under a comb graph

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## Abstract

We consider a linear time–cost tradeoff problem (LTCTP) with multiple milestones under a comb graph. A time–cost tradeoff problem decides whether and how much to spend a cost to compress processing time of a job in order to meet promised due dates. This is common in managing a project because additional labors or resources are expensive. It is called linear where the compression cost linearly increases with reduced time. An objective of the LTCTP that this study addresses is to minimize the weighted number of tardy milestones plus the total compression cost. This study considers a special precedence structure, which is called a comb graph. A chain of jobs that sequentially precede each other forms a main process of a project, and other chains of jobs, corresponding to sub processes, precede each job in the main process. For this structure, we developed a strongly-polynomial-time algorithm. This is the first result of unveiling complexity of the multi-milestone LTCTP under a non-chain structure.

## Keywords

Project scheduling Time–cost tradeoff Multiple milestones Comb graph## Notes

### Acknowledgements

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2018S1A5B8070344).

## References

- Ahuja RA, Mehlhorn K, Orlin JB (1990) Faster algorithm for the shortest path problem. J Assoc Comput Mach 37:213–223MathSciNetCrossRefzbMATHGoogle Scholar
- Artigues C, Demassey S, Nerson E (2008) Resource-constrained project scheduling. Wiley, New YorkCrossRefGoogle Scholar
- Bruker P, Drexl A, Mohring R, Neumann K, Perch E (1999) Resource-constrained project scheduling: notations, classfication, models and methods. Eur J Oper Res 112:3–41CrossRefGoogle Scholar
- Choi BC, Chung JB (2013) Project time–cost tradeoff problem with milestones under an uncertain processing time. J Korean Oper Res Manag Sci Soc 38:25–42Google Scholar
- Choi BC, Chung JB (2014) Complexity results for the linear time–cost tradeoff problem with multiple milestones and completely ordered jobs. Eur J Oper Res 236:61–68MathSciNetCrossRefzbMATHGoogle Scholar
- Choi BC, Chung JB (2016) Some special cases of a continuous time–cost tradeoff problem with multiple milestones under a chain precedence graph. Manag Sci Financ Eng 22:5–12CrossRefGoogle Scholar
- Choi BC, Park MJ (2015) A continuous time-cost tradeoff problem with multiple milestones and completely ordered jobs. Eur J Oper Res 244:748–752MathSciNetCrossRefzbMATHGoogle Scholar
- De P, Dunne J, Ghosh JB, Wells CE (1997) Complexity of the discrete time–cost trade-off problem for project networks. Oper Res 45:302–306MathSciNetCrossRefzbMATHGoogle Scholar
- Deckstein D (2011) Jumbo problems: Dreamliner becomes a nightmare for Boeing. Spiegel Online, March 30Google Scholar
- Demeulemeester EL, Herroelen WS (2002) Project scheduling—a research handbook. Kluwer Academic, BostonzbMATHGoogle Scholar
- Fulkerson DR (1961) A network flow computation for project cost curves. Manag Sci 7:167–178MathSciNetCrossRefzbMATHGoogle Scholar
- Hillier FS, Lieberman GJ (2001) Introduction to operations research. McGraw-Hill, New YorkzbMATHGoogle Scholar
- Kelley JE (1961) Critical path planning and scheduling: mathematical basis. Oper Res 9:296–320MathSciNetCrossRefzbMATHGoogle Scholar
- Shtub A, Bard JF, Globerson S (1994) Project management: engineering, technology and implementation. Prentice Hall, Englewood CliffGoogle Scholar
- Skutella M (1998) Approximation algorithms for the discrete time–cost tradeoff problem. Math Oper Res 23:909–929MathSciNetCrossRefzbMATHGoogle Scholar
- Ulrich K, Eppinger S (2011) Product design and development. McGraw-Hill/Irwin, New YorkGoogle Scholar
- Weglarz J (1999) Project scheduling—recent models algorithms and applications. Kluwer Academic, BostonGoogle Scholar
- Weglarz J, Jozefowska J, Mika M, Waligora G (2011) Project scheduling with finite or infinite number of activity processing modes—a survey. Eur J Oper Res 208:177–205MathSciNetCrossRefzbMATHGoogle Scholar