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An adaptive genetic algorithm for the time dependent inventory routing problem

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Abstract

In this paper we propose an adaptive genetic algorithm that produces good quality solutions to the time dependent inventory routing problem (TDIRP) in which inventory control and time dependent vehicle routing decisions for a set of retailers are made simultaneously over a specific planning horizon. This work is motivated by the effect of dynamic traffic conditions in an urban context and the resulting inventory and transportation costs. We provide a mixed integer programming formulation for TDIRP. Since finding the optimal solutions for TDIRP is a NP-hard problem, an adaptive genetic algorithm is applied. We develop new genetic representation and design suitable crossover and mutation operators for the improvement phase. We use adaptive genetic operator proposed by Yun and Gen (Fuzzy Optim Decis Mak 2(2):161–175, 2003) for the automatic setting of the genetic parameter values. The comparison of results shows the significance of the designed AGA and demonstrates the capability of reaching solutions within 0.5 % of the optimum on sets of test problems.

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Abbreviations

\(N\) :

Number of retailers including depot

\(T\) :

Number of time periods

\(K\) :

Number of available vehicles during each period

\(M\) :

Number of time intervals considered for each link

\(t\) :

The starting time from the depot node 1

\(b_k \) :

Volume capacity of vehicle \(k\)

\(C_i \) :

Retailer’s own capacity to hold inventory

\(d_{it} \) :

Amount of demand at node \(i\) during period \(t\)

\(c_{ij}^{tm} \) :

Travel time from node \(i\) to \(j\) if starting at \(i\) during time interval \(m\) at period \(t\); \(c_{ii}^{tm} =\infty \) for all \(i, m\)

\(s_{it} \) :

Service time at node \(i\) at period \(t\)

\(T_{ij}^{tm} \) :

Upper bound for time interval \(m\) for link \(\left( {i, j} \right)\) at period \(t\)

\(h_i^+ \) :

Holding cost per unit at node \(i\)

\(h_i^- \) :

Backorder cost per unit at node \(i\)

\(c\) :

Variable routing cost per hour

\(f_t \) :

Fixed cost per vehicle at period \(t\)

\(B\) :

Max\(_{k}b_k = \) capacity of largest vehicle

\(x_{ij}^{tm}\) :

\(\left\{ {{\begin{array}{ll} 1&\quad \text{ if} \text{ any} \text{ vehicle} \text{ travels} \text{ directly} \text{ from} \text{ node}\\&\quad i \text{ to} \text{ node} j \text{ starting} \text{ from} i \text{ during} \text{ time}\\&\quad \text{ interval} m \text{ at} \text{ period} t \\ 0&\quad \text{ otherwise} \\ \end{array} }} \right.\)

\(y_{ij}^{tm} \) :

Amount transported on that trip for time interval \(m\) for link \(\left( {i, j} \right)\) at period \(t\)

\(I_{it}\) :

Inventory levels at node \(i\) at period \(t\)

\(B_{it}\) :

Inventory stock-outs at node \(i\) at period \(t\)

\(t_{it}\) :

Departure time of any vehicle from node \(i\) at period \(t\)

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Acknowledgments

This work was supported by the research fund of Hanyang University (HY-2011-P).

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Authors and Affiliations

  1. Department of Industrial and Management Engineering, Hanyang University, Ansan, Gyeonggi-do, 426-791, South Korea

    Dong Won Cho, Young Hae Lee & Tae Youn Lee

  2. Fuzzy Logic Systems Institute, Iizuka, Japan

    Mitsuo Gen

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  1. Dong Won Cho
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  2. Young Hae Lee
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  3. Tae Youn Lee
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Correspondence to Young Hae Lee.

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Cho, D.W., Lee, Y.H., Lee, T.Y. et al. An adaptive genetic algorithm for the time dependent inventory routing problem. J Intell Manuf 25, 1025–1042 (2014). https://doi.org/10.1007/s10845-012-0727-5

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  • DOI: https://doi.org/10.1007/s10845-012-0727-5

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