# A unified tabu search algorithm for vehicle routing problems with soft time windows

## Summary (3 min read)

### Introduction

- The basic vehicle routing problem (VRP) calls for the determination of a set of minimum-cost routes to be performed by a fleet of vehicles to serve a given set of customers with known demands, where each route originates and terminates at a single depot.
- Therefore, the priority is given to the first objective.
- The vehicle routing problem with time windows is an extension of the basic VRP in which vehicle capacity constraints are imposed and each customer i is associated with a time interval [ai , bi ], called a time window, during which service must begin.
- In any vehicle route, the vehicle may not arrive at customer i after bi to begin service.
- (3) Setting the time windows to be soft may allow significant savings in the number of vehicles required and/or the total vehicle travel distance of all routes to be achieved.

### Notation and problem representation

- If any edges between vertices are missing in the original graph, then they are replaced by edges with an artificially high cost.
- The start time of each vehicle route is greater than or equal to a0.
- In practice, these coefficients could represent the costs of lost sales, goodwill, etc, due to the customer inconvenience for not meeting the time windows, and are therefore often called inconvenience costs.
- Chiang and Russell (2004) developed a TS solution method for the problem.
- Based on Type 2, define Pmax as the maximum allowable violation of the time windows, also known as Type 5.

### The tabu search algorithm

- The algorithm is based on tabu search, a local search metaheuristic introduced first by Fred Glover in 1986, and since then has been used to solve many practical applications.
- A thorough discussion may be found in Glover and Laguna (1997).
- The authors developed an effective TS heuristic for the open vehicle routing problem (OVRP) in a previous work (Fu et al, 2005, 2006).
- The full details of the algorithm were given in the referenced paper and so only the outline of the algorithm and the changes that have been made to apply it to the VRPSTW are described here.

### Initial solution

- An initial solution is required for any TS algorithm to start the local search process.
- With respect to the TS metaheuristic for the VRP, some researchers, such as Van Breedam (2001) and Brandão (2004), claimed that the performance of the TS heuristic was highly dependent on the quality of the initial solutions.
- In their previous work (Fu et al, 2005), it was shown that the initial solution did not have a large influence on the final solutions in the proposed TS heuristic.
- Therefore, the initial solution is generated by building up successive routes where the next customer is chosen at random and added to the end of the route unless this violates the capacity or route length constraints; in that case the route is completed back to the depot and a new route starts.

### Neighbourhood structure

- The TS algorithm presented in this paper uses the same mixed neighbourhood structure as was introduced in Fu et al (2005), but some necessary modification is made for the VRPSTW where each vehicle route must start and end at the depot.
- There are four different types of move that may be used in this mixed neighbourhood structure, and the algorithm selects a type of move randomly at each iteration.
- The four different types of move are referred to as vertex reassignment, vertex swap, 2-opt and ‘tails’ swap.
- The first and the last 0 of a solution are not allowed to be selected and removed during this 2-interchange process.

### Evaluation of solutions

- The extent of infeasibility can be measured by incorporating the vehicle capacity and maximum route length constraints into the objective function by adding a penalty if the constraints are broken.
- The tabu list and stopping criterion also follow the method described in Fu et al (2005).
- The tabu list contains the move attributes of solutions during the last 5–10 (selected randomly) iterations.
- The search is terminated if either a specified number of iterations have elapsed in total or since the last best solution was found.

### TS algorithm

- The unified TS algorithm for VRPSTW is described below.
- Generate an initial feasible solution randomly, and set this solution as the current solution and the best solution so far;.
- Set the new solution as the current solution, update the tabu list and increment iter;.
- Its simple but powerful mixed neighbourhood structure and the stochastic elements in the method allow effective diversification within a local search framework.

### Computational results and comparison

- This unified TS algorithm was coded in Delphi 7.0 and implemented on a Pentium-II PC running on 600MHz with 184MB RAM.
- In problem sets R1, C1 and RC1, the time window is narrow at the central depot so that only a few customers can be served on each route.
- The authors unified TS algorithm for three of the six main types of VRPSTW that were previously defined (Type 1, Type 2, and Type 3) was tested on the benchmark problems in the literature.

### Comparison of the results on Type 1 of VRPSTW

- The results reported were feasible solutions to the VRPHTW in each case, that is, the percentage of non-violated time windows was 100%, and the number of routes was set to the best solution reported in the literature for each problem, not minimized by the algorithm.
- The real Euclidean distances between customers are used during the computations, whereas the final results are rounded to the second decimal.
- In the table, their best solutions are compared with those produced by Taillard et al (1997) for the VRPSTW and other algorithms reported in the literature for the VRPHTW, respectively, using the format: number of routes/total travel distance.
- When the percentage of non-violated time windows is 100%, they are compared with those for the VRPHTW as well.
- An ‘H’ after ∗∗ indicates that their algorithm has improved the best published solution and after ∗ means a tie with the best published solution for the VRPHTW.

### Comparison of the results on Type 2 of VRPSTW

- The heuristic started with low penalty coefficients, which were gradually increased.
- The comparison of the results is shown in Table 3, using the format: number of routes/total travel distance, percentage of non-violated time windows.
- Among the 12 improved solutions, eight of them require a lower number of vehicle routes and four are of shorter total travel distance (for the same number of routes required and non-violated time windows).
- For the problem sets of C1, comparing with the heuristic of Koskosidis et al (1992) as well as their algorithm for Type 1 of VRPSTW, their algorithm for Type 2 of VRPSTW takes much more CPU time to find the same best known solutions, and cannot obtain the best known solutions in two instances.

### Comparison of the results on Type 3 of VRPSTW

- For Type 3 of VRPSTW, Balakrishnan (1993) described three simple heuristics, Chiang and Russell (2004) developed a TS solution method.
- The eight problems based on the R1 and RC1 sets of the Solomon benchmark problems were used to test the algorithms.
- The hard time windows in the original benchmark data were converted to soft time windows by allowing a certain percentage of time window violation, Emax = Lmax, that is, Pmax, as in Balakrishnan (1993) and Chiang and Russell (2004).
- Pmax (Emax and Lmax) and Wmax were expressed as a percentage of the maximum allowable route time duration.

### Conclusions

- The different forms of time window violation allowed lead to different types of VRPSTW.
- The existing approaches in the literature are usually designed for a special type of VRPSTW.
- Finally, to test the computational performance of the algorithm, the authors ran it on the benchmark problems and compared the results with other methods for three types of VRPSTW in the literature.
- Acknowledgements—We are grateful to the anonymous referees for their useful comments and suggestions that helped us to improve the presentation of this paper.the authors.the authors.

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328 citations

### Cites background from "A unified tabu search algorithm for..."

...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al. (2011) PR08 Pirkwieser and Raidl (2008) ZK11 Zachariadis and Kiranoudis (2011) HDH09 Hemmelmayr et al. (2009) RP06 Ropke and Pisinger (2006a) ZK12 Zachariadis and Kiranoudis (2012) ISW09 Imran et al. (2009) RL12 Ribeiro and Laporte (2012)...

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...State-of-the-art algorithms: B10 Belhaiza (2010) KTDHS12 Kritzinger et al. (2012) RT10 Repoussis and Tarantilis (2010) BDHMG08 Bräysy et al. (2008) MB07 Mester and Bräysy (2007) RTBI10 Repoussis et al. (2010) BER11 Bektas et al. (2011) MCR12 Moccia et al. (2012) RTI09a Repoussis et al. (2009a) BLR11 Balseiro et al. (2011) NB09 Nagata and Bräysy (2009) RTI09b Repoussis et al. (2009b) BPDRT09 Bräysy et al. (2009) NBD10 Nagata et al. (2010) SDBOF10 Subramanian et al. (2010) CM12 Cordeau and M. (2012) NPW10 Ngueveu et al. (2010) SPUO12 Subramanian et al. (2012) F10 Figliozzi (2010) P09 Prins (2009) SUO13 Subramanian et al. (2013) FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al. (2011) PR08 Pirkwieser and Raidl (2008) ZK11 Zachariadis and Kiranoudis (2011) HDH09 Hemmelmayr et al. (2009) RP06 Ropke and Pisinger (2006a) ZK12 Zachariadis and Kiranoudis (2012) ISW09 Imran et al. (2009) RL12 Ribeiro and Laporte (2012) Table 3 displays the list of acronyms for the benchmark instances and methods used in the comparative analysis....

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...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al. (2011) PR08 Pirkwieser and Raidl (2008) ZK11 Zachariadis and Kiranoudis (2011) HDH09 Hemmelmayr et al. (2009) RP06 Ropke and Pisinger (2006a) ZK12 Zachariadis and Kiranoudis (2012) ISW09 Imran et al....

[...]

...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al. (2010) ZTK10 Zachariadis et al. (2010) GA09b Gajpal and Abad (2009a) PR07 Pisinger and Ropke (2007) ZK10 Zachariadis and Kiranoudis (2010) GG11 Groër et al....

[...]

...FEL07 Fu et al. (2007) PBDH08 Polacek et al. (2008) XZKX12 Xiao et al. (2012) GA09a Gajpal and Abad (2009b) PDDR10 Prescott-Gagnon et al....

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165 citations

145 citations

133 citations

### Cites background or methods from "A unified tabu search algorithm for..."

...Hashimoto et al. (2006) proposes an algorithm for flexible time windows (hard and soft) and travel times using local search; soft time window and soft traveling time constraints are treated as part of the objective function and the authors deal with a generalized VRP. Fu et al. (2008) adapted a tabu search algorithm, previously used in the open vehicle routing problem (Fu et al., 2005), for the VRPSTW....

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...Construction heuristics include the work of Solomon (1987), Potvin and Rousseau (1993), and Ioannou et al. (2001)....

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...The solution results presented by Fu et al. (2008) using the unified tabu search method are denoted (UTS)....

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...Balakrishnan (1993), Chiang and Russell (2004), and Fu et al. (2008) also set a maximum vehicle waiting time limit Wmax....

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...To the best of the author’s knowledge, the only three journal publications that include results for VRPSTW benchmark problems and comply with prerequisites (a)–(c) are: Balakrishnan (1993), Chiang and Russell (2004), and Fu et al. (2008)....

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93 citations

### Cites background from "A unified tabu search algorithm for..."

...3(ei − bi) as in [47], where [bi, ei] denotes the time window of i....

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##### References

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### "A unified tabu search algorithm for..." refers background or result in this paper

...lems generated by Solomon (1987) (can be downloaded from: http://w....

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...To show the computational performance of the algorithm, it was tested on the 56 benchmark problems generated by Solomon (1987) (can be downloaded from: http://w.cba.neu.edu/∼msolomon/problems.htm) and the results were compared with others in the literature....

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...Koskosidis et al (1992) developed an optimization-based heuristic for Type 2 of VRPSTW and tested it on the five sets of randomly generated problems and 21 instances of 56 Solomon’s benchmark problems. The heuristic started with low penalty coefficients, which were gradually increased. In our algorithm, we suppose the penalty coefficients i = i = 100 for all i and run it for the 21 instances listed. The comparison of the results is shown in Table 3, using the format: number of routes/total travel distance, percentage of non-violated time windows. The comparison of the CPU time in seconds is in Table 4. Our algorithm has improved 12 and tied seven solutions on the 21 instances listed, indicated by ∗∗ and ∗, respectively. Among the 12 improved solutions, eight of them require a lower number of vehicle routes and four are of shorter total travel distance (for the same number of routes required and non-violated time windows). For the problem sets of C1, comparing with the heuristic of Koskosidis et al (1992) as well as our algorithm for Type 1 of VRPSTW, our algorithm for Type 2 of VRPSTW takes much more CPU time to find the same best known solutions, and cannot obtain the best known solutions in two instances....

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