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Designing a multi-depot multi-period vehicle routing problem with time window: hybridization of tabu search and variable neighbourhood search algorithm

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Abstract

This research formulates a real-life multi-depot and multi-period vehicle routing problem (MDMPVRP) by imposing time window (TW) and many other constraints. A set of customers, spread in different locations, are to be served by a fleet of heterogeneous vehicles over a finite number of periods. Each customer is associated with combinations of routes and vehicles over the period. A customer must be served in one of the allowable combinations. The objective of this MDMPVRP-TW is to minimize the total distance traversed by the fleet over the planning horizon. The proposed MDMPVRP-TW is an extension of vehicle routing problem (VRP), and is hence an NP-hard problem. In order to optimize it, we propose a hybrid meta-heuristic approach by combining tabu search (TS) and variable neighbourhood search (VNS) algorithms. Furthermore, to provide richer insights, the efficacy of the proposed method and mathematical formulation is demonstrated through numerical experiments for a number of instances varying from small to large scale.

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Abbreviations

\( V_1=\{1,2,\ldots ,n\} \) :

Set of customers

\( V_2=\{n+1,\ldots ,n+d\} \) :

Set of depots

\( V=V_1\cup V_2 \) :

Set of all nodes

i, j :

Indices for the customer nodes over the set V

t :

Index for the time period over the set \(T=\{1,2,\ldots ,p\}\)

k :

Index for heterogeneous vehicle over the set \(K=\{1,2,\ldots ,r\}\)

\( D_k \) :

Maximum duration of vehicle k

\( Q_k \) :

Capacity of the vehicle k

v :

Average vehicle speed

\( r_i \) :

Number of allowable combinations of ith customer

\( q_{it}\) :

Demand at ith node in period t

\( d_{ij}\) :

Distance between the nodes i and j

[\(e_{it},l_{it}\)]:

Time interval in which service must start at node i in period t

\( s_{ikt} \) :

Service time required at i in tth period by kth vehicle

\( f_i \) :

Visit frequency to the node i over entire planning horizon

\( C_i \) :

Set of visit combinations of customer i

\( a_{rt} \) :

1 if day t belongs to combination r, otherwise 0

\( \Lambda \) :

\(\max \) \(\{l_{it}|i\in V_2\}\)

\( ar_{ikt} \) :

Arrival time of the vehicle k at i in tth period

\( y_{ikt} \) :

Remaining cargo of kth vehicle on arrival at i in tth period

\( w_{ir} \) :

1 if visit combination \(r\in C_i\) is selected for node i, otherwise 0

\( E_{ikt} \) :

1 if vehicle k visits depot i in period t for intermediate replenishment, otherwise 0

\( M_{ikt} \) :

1 if vehicle k starts a tour from ith depot in period t, otherwise 0

\( N_{ikt} \) :

1 if vehicle k ends a tour at ith depot in period t, otherwise 0

\( x_{ijkt} \) :

1 if vehicle k goes from i to j in period t, otherwise 0

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Acknowledgements

The authors would like to acknowledge the support provided by the Indian Institute of Technology Kharagpur through facilities for research. The first and third authors are grateful to Ministry of Human Resource Development for supporting their scientific studies with the Institute Research Assistantship.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Arjun Paul, Chayanika Rout & Adrijit Goswami

  2. Government Polytechnic, Saharsa, DST Bihar, Bihar, 852201, India

    Ravi Shankar Kumar

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  1. Arjun Paul
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  2. Ravi Shankar Kumar
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  3. Chayanika Rout
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  4. Adrijit Goswami
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Correspondence to Adrijit Goswami.

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Paul, A., Kumar, R.S., Rout, C. et al. Designing a multi-depot multi-period vehicle routing problem with time window: hybridization of tabu search and variable neighbourhood search algorithm. Sādhanā 46, 183 (2021). https://doi.org/10.1007/s12046-021-01693-2

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