# Chinese and windy postman problem with variable service costs

- 147 Downloads

## Abstract

Given a network \({\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})\) of nodes denoted by \({\mathcal {V}}\), edges between nodes represented by \({\mathcal {E}}\), and costs associated with the edges, postman problem (PP) is to find the route having the minimum cost that begins and ends with a predefined starting point and spans each edge of the network. PP is a variant of the well-known arc routing problem. In many real-life applications of the PP, costs associated with the edges tend to reduce with each pass on the edges. We propose a new mathematical formulation to represent the postman problem with variable service costs. If the service costs are symmetric, the problem is named as the Chinese postman problem (CPP) with variable service costs (CPPVSC), and it is called as the windy postman problem with variable service costs (WPPVSC), otherwise. CPPVSC turns to be a variant of CPP, and it is an easy problem. We show that no edge can be traversed more than twice in the optimal solution. Moreover, we propose two heuristics for the solution of WPPVSC. Based on the extensive numerical experiments, we can say that both heuristics outperform the state-of-the-art commercial solvers.

## Keywords

Chinese postman problem Windy postman problem Variable service costs Heuristic approaches## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- Assad AA, Golden BL (1995) Arc routing methods and applications. Handb Oper Res Manag Sci 8:375–483MathSciNetzbMATHGoogle Scholar
- Ávila T, Corberán Á, Plana I, Sanchis JM (2016) A branch-and-cut algorithm for the profitable windy rural postman problem. Eur J Oper Res 249(3):1092–1101MathSciNetCrossRefzbMATHGoogle Scholar
- Benavent E, Campos V, Corberán A, Mota E (1992) The capacitated arc routing problem: lower bounds. Networks 22(7):669–690MathSciNetCrossRefzbMATHGoogle Scholar
- Black D, Eglese R, Wøhlk S (2013) The time-dependent prize-collecting arc routing problem. Comput Oper Res 40(2):526–535MathSciNetCrossRefzbMATHGoogle Scholar
- Campbell JF, Langevin A (2000) Roadway snow and ice control. In: Dror M (ed) Arc routing. Springer, Berlin, pp 389–418Google Scholar
- Cordeau JF, Ghiani G, Guerriero E (2012) Analysis and branch-and-cut algorithm for the time-dependent travelling salesman problem. Transp Sci 48(1):46–58CrossRefGoogle Scholar
- Donati AV, Montemanni R, Casagrande N, Rizzoli AE, Gambardella LM (2008) Time dependent vehicle routing problem with a multi ant colony system. Eur J Oper Res 185(3):1174–1191MathSciNetCrossRefzbMATHGoogle Scholar
- Dror M (2012) Arc routing: theory, solutions and applications. Springer, DordrechtGoogle Scholar
- Dussault B, Golden B, Groër C, Wasil E (2013) Plowing with precedence: a variant of the windy postman problem. Comput Oper Res 40(4):1047–1059MathSciNetCrossRefzbMATHGoogle Scholar
- Eglese R, Li L (1992) Efficient routeing for winter gritting. J Oper Res Soc 43(11):1031–1034CrossRefGoogle Scholar
- Gendreau M, Ghiani G, Guerriero E (2015) Time-dependent routing problems: a review. Comput Oper Res 64:189–197MathSciNetCrossRefzbMATHGoogle Scholar
- Golden BL, DeArmon JS, Baker EK (1983) Computational experiments with algorithms for a class of routing problems. Comput Oper Res 10(1):47–59MathSciNetCrossRefGoogle Scholar
- Guan M (1984) On the windy postman problem. Discrete Appl Math 9(1):41–46MathSciNetCrossRefzbMATHGoogle Scholar
- Gurobi optimizer 6.0.: high-end libraries for math programming (2017). http://www.gurobi.com/. Accessed Jan 2017
- Hertz A (2005) Recent trends in arc routing. In: Golumbic MC, Hartman IBA (eds) Graph theory, combinatorics and algorithms. Springer, Berlin, pp 215–236Google Scholar
- Ichoua S, Gendreau M, Potvin JY (2003) Vehicle dispatching with time-dependent travel times. Eur J Oper Res 144(2):379–396CrossRefzbMATHGoogle Scholar
- Koç Ç, Bektaş T, Jabali O, Laporte G (2016) Thirty years of heterogeneous vehicle routing. Eur J Oper Res 249(1):1–21MathSciNetCrossRefzbMATHGoogle Scholar
- Li LY, Eglese RW (1996) An interactive algorithm for vehicle routeing for winter-gritting. J Oper Res Soc 47(2):217–228Google Scholar
- Li F, Golden B, Wasil E (2005) Solving the time dependent traveling salesman problem. In: Golden B, Raghavan S (eds) The next wave in computing, optimization, and decision technologies. Springer, Berlin, pp 163–182Google Scholar
- Malandraki C, Daskin MS (1992) Time dependent vehicle routing problems: formulations, properties and heuristic algorithms. Transp Sci 26(3):185–200CrossRefzbMATHGoogle Scholar
- Malandraki C, Dial RB (1996) A restricted dynamic programming heuristic algorithm for the time dependent traveling salesman problem. Eur J Oper Res 90(1):45–55CrossRefzbMATHGoogle Scholar
- Schneider J (2002) The time-dependent traveling salesman problem. Phys A Stat Mech Appl 314(1):151–155MathSciNetCrossRefzbMATHGoogle Scholar
- Setak M, Habibi M, Karimi H, Abedzadeh M (2015) A time-dependent vehicle routing problem in multigraph with fifo property. J Manuf Syst 35:37–45CrossRefGoogle Scholar
- Sun J, Meng Y, Tan G (2015) An integer programming approach for the chinese postman problem with time-dependent travel time. J Comb Optim 29(3):565–588MathSciNetCrossRefzbMATHGoogle Scholar
- Tagmouti M, Gendreau M, Potvin JY (2007) Arc routing problems with time-dependent service costs. Eur J Oper Res 181(1):30–39MathSciNetCrossRefzbMATHGoogle Scholar
- Tagmouti M, Gendreau M, Potvin JY (2010) A variable neighborhood descent heuristic for arc routing problems with time-dependent service costs. Comput Ind Eng 59(4):954–963CrossRefGoogle Scholar
- Tagmouti M, Gendreau M, Potvin JY (2011) A dynamic capacitated arc routing problem with time-dependent service costs. Transp Res Part C Emerg Technol 19(1):20–28CrossRefzbMATHGoogle Scholar
- Taş D, Gendreau M, Jabali O, Laporte G (2016) The traveling salesman problem with time-dependent service times. Eur J Oper Res 248(2):372–383MathSciNetCrossRefzbMATHGoogle Scholar
- Test instances for arc routing problems. http://www.uv.es/corberan/instancias.htm (2017). Accessed Jan 2017
- Vincent FY, Lin SW (2015) Iterated greedy heuristic for the time-dependent prize-collecting arc routing problem. Comput Ind Eng 90:54–66CrossRefGoogle Scholar