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A lagrangian relaxation algorithm for facility location of resource-constrained decentralized multi-project scheduling problems

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Abstract

Recent literature on multi-project scheduling problems has been focused on the transfer of available resources between project activities, especially in decentralized projects. Determining the right location for storage facilities where resources are moved between activities is major topics in decentralized multi-projects scheduling problems. This paper introduces a new decentralized resource-constrained multi-project scheduling problem. Its purpose is to simultaneously minimize the cost of the project completion time and the cost of facilities location. A mixed integer linear programming model is first presented to solve small-size problems by using conventional solvers. Then, three heuristic/meta-heuristic methods are proposed to solve larger-size problems. To this end, a fast constructive heuristic algorithm based on priority rules is introduced. Then, using the heuristic method structure, a combinatorial genetic algorithm is developed to solve large-size problems in reasonable CPU running time. Finally, the lagrangian relaxation technique and branch-and-bound algorithm are applied to generate an effective lower bound. According to the computational results obtained and managerial insights, total costs can be significantly reduced by selecting an optimal location of resources. By the use of a scenario-based TOPSIS approach, the heuristic methods are ranked based on changes in the importance of metrics. Friedman’s test result of TOPSIS shows that there is no significant difference among the heuristic methods.

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Science and Technology (IUST), Tehran, Iran

    Mohammad Rostami & Morteza Bagherpour

Authors
  1. Mohammad Rostami
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  2. Morteza Bagherpour
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Corresponding author

Correspondence to Morteza Bagherpour.

Appendix A

Appendix A

The steps of TOPSIS approach for ranking heuristic methods:

  • Decision matrix is created

  • The elements of this matrix are normalized based on Eq. (29):

$$f_{ij} = \frac{{m_{ij} }}{{\sqrt {\sum\nolimits_{i = 1}^{\gamma } {m_{ij}^{2} } } }}$$
(29)

where, \(m_{ij}\) is the element of decision matrix and \(f_{ij}\) is normalized value. \(\gamma\) is the number of heuristic methods.

  • With the help of weights specified for the sub metrics in each scenario, the weighted normalized matrix is calculated.

  • In each sub metric, the best (\(f_{bj}\)) and the worst (\(f_{wj}\)) element is specified.

  • With the help of Eqs. (30) and (31), the Euclidean distances of heuristic methods from the best and the worst is calculated:

$$db_{i} = \sqrt {\sum\limits_{j = 1}^{\varphi } {\left( {f_{ij} - f_{bj} } \right)} }$$
(30)
$$dw_{i} = \sqrt {\sum\limits_{j = 1}^{\varphi } {\left( {f_{ij} - f_{wj} } \right)} } \,$$
(31)

where, \(\varphi\) shows the number of sub metrics.

  • Finally, the criteria similar to worst (\(SW_{i}\)) is calculated by Eq. (32) and the method with the highest \(SW_{i}\) is selected as the best.

$$SW_{i} = \frac{{dw_{i} }}{{\left( {dw_{i} + db_{i} } \right)}}$$
(32)

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Rostami, M., Bagherpour, M. A lagrangian relaxation algorithm for facility location of resource-constrained decentralized multi-project scheduling problems. Oper Res Int J 20, 857–897 (2020). https://doi.org/10.1007/s12351-017-0358-x

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