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Penalty Rules in Multicriteria Genetic Search

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New State of MCDM in the 21st Century

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 648))

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Abstract

In the paper, by means of numerical experiments conducted on artificially constructed problem instances, we test penalty rules for constrained genetic optimization of the Capacitated Heterogeneous Vehicle Routing Problem with Time-Windows in a bi-objective framework. Optimized criteria are cost minimization and capacity utilization maximization. Two approaches are employed – scalarization of objectives and dominance-based evaluation of solutions. We show that it is possible to handle infeasibility in such a way, that this risk of divergence to regions of infeasibility is acceptable. The most secure penalty rule among the tested ones turns out to be the rule which explicitly controls the proportion of infeasible solutions in the population. This rule, along with the rule which accounts only the notion of solutions distance from the feasible set, outperforms rules based on time-penalties and best to best-feasible solution comparison over considered case studies.

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Notes

  1. 1.

    These extensions include, among others, the Vector Evaluated GA (Schaffer 1985), Multiple-Objective GA (Fonseca and Fleming 1993), (Narayanan and Azarm 1999) and the Nondominated Sorting GA (Srinivas and Deb 1994).

  2. 2.

    A survey of constraints handling methods is e.g. Coello (1999).

  3. 3.

    Real life Multiple Criteria Capacitated Vehicle Routing Problem with Time Windows which is considered in this paper can serve as an example.

  4. 4.

    For a survey on multiple criteria VRPs, see Jozefowicz et al. (2008). For some recent applications of genetic algorithms in the domain of multiobjective VRPs see e.g. Chand et al. (2007) or Tan and Chew (2006).

  5. 5.

    Standard perception of VRPs is that they help practitioners to construct an optimal transportation plan given available fleet of vehicles. In this respect capacity utilization maximization is purposeful, for it helps reducing the cost per unit of transported goods. On the other hand, this criterion can be exploited when one wants to answer the question what a fleet should be constructed to best serve existing needs. The optimization routine indicates which vehicles should be used, and maximization of capacity utilization helps composing a fleet that minimizes waste of free space in containers.

  6. 6.

    Elements of α are nonnegative and sum up to unity. In the numerical simulations section we perform sensitivity analysis of results with respect to α. In practice elements of α have to be elicited from the decision maker.

  7. 7.

    A random number of possible insertion places is considered and the best one is chosen.

  8. 8.

    Due to the fact that some of its orders were erased or due to the fact that sequence s1 or s2 was inserted in them. We use the rule, that a vehicle with minimal capacity, but able to handle a route is assigned to it.

  9. 9.

    There is at least one route in σ, the one that starts in the first element of σ.

  10. 10.

    These characteristics are: topology of the transportation network, distribution of clients locations over the network, number of orders placed by each client, volume of goods ordered within each order, time windows of orders and sizes of available vehicle types.

  11. 11.

    These probability distributions were uniform over appropriate intervals.

  12. 12.

    For rules (4) and (5) this characteristic is not reported, for the fraction of infeasible solutions is explicitly controlled for.

  13. 13.

    This was the case also for other rules for which results are not reported here.

  14. 14.

    Results were similar for these two approaches.

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Authors and Affiliations

  1. Warsaw School of Economics, Al. Niepodleglosci 164, 02-554, Warsaw, Poland

    Grzegorz Koloch & Tomasz Szapiro

Authors
  1. Grzegorz Koloch
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  2. Tomasz Szapiro
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Correspondence to Grzegorz Koloch .

Editor information

Editors and Affiliations

  1. Research Cntr on Fict. Econ. & Data Sci., Chinese Academy of Sciences, Zhongguancun East Road, No 80 Building 6, Room 203, Beijing, 10090, China, People's Republic

    Yong Shi

  2. College of Information Science and Technology, University of Nebraska at Omaha, Omaha, NE 68182, USA

    Yong Shi

  3. Academy of Mathematics and, Chinese Academy of Sciences, Zhonguancun East Road 55Haidian, Beijing, NE, 100190, China, People's Republic

    Shouyang Wang

  4. School of Management and Economics, University of Electr. Science & Technol., Zhonguancun East Road No. 55, Haidian, Chengdu, 610054, China, People's Republic

    Gang Kou PhD

  5. Aalto University School of Economics, 21210, 00076, Aalto, Finland

    Jyrki Wallenius

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Koloch, G., Szapiro, T. (2011). Penalty Rules in Multicriteria Genetic Search. In: Shi, Y., Wang, S., Kou, G., Wallenius, J. (eds) New State of MCDM in the 21st Century. Lecture Notes in Economics and Mathematical Systems, vol 648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19695-9_8

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