Soft Computing

, Volume 23, Issue 10, pp 3279–3301 | Cite as

Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint

  • Saibal Majumder
  • Pradip Kundu
  • Samarjit KarEmail author
  • Tandra Pal
Methodologies and Application


Modeling of real-world problems requires data as input parameter which include information represented in the state of indeterminacy. To deal with such indeterminacy, use of uncertainty theory (Liu in Uncertainty theory, Springer, Berlin, 2007) has become an important tool for modeling real-life decision-making problems. This study presents a profit maximization and time minimization scheme which considers the existence of possible indeterminacy by designing an uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint (UMMFSTPwB) at each destination. Here, items are purchased at different source points with different prices and are accordingly transported to different destinations using different types of vehicles. The items are sold to the customers at different selling prices. In the proposed model, unit transportation costs, fixed charges, transportation times, supplies at origins, demands at destinations, conveyance capacities and budget at destinations are assumed to be uncertain variables. To model the proposed UMMFSTPwB, we have developed three different models: (1) expected value model, (2) chance-constrained model and (3) dependent chance-constrained model using uncertain programming techniques. These models are formulated under the framework of uncertainty theory. Subsequently, the equivalent deterministic transformations of these models are formulated and are solved using three different methods: (1) linear weighted method, (2) global criterion method and (3) fuzzy programming method. Finally, numerical examples are presented to illustrate the models.


Multi-objective solid transportation problem Profit maximization Fixed charge Budget constraint Uncertain programming 



The authors are deeply indebted to the Editor and the anonymous referees for their constructive and valuable suggestions to enhance the quality of the manuscript. Moreover, Saibal Majumder, an INSPIRE fellow (No. DST/INSPIRE Fellowship/2015/IF150410) would like to acknowledge Department of Science & Technology (DST), Government of India, for providing him financial support for the work.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Ethical approval

This article does not contains any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Saibal Majumder
    • 1
  • Pradip Kundu
    • 2
  • Samarjit Kar
    • 3
    Email author
  • Tandra Pal
    • 1
  1. 1.Department of Computer Science and EngineeringNational Institute of TechnologyDurgapurIndia
  2. 2.Department of Mathematics and StatisticsIndian Institute of Science Education and Research KolkataMohanpurIndia
  3. 3.Department of MathematicsNational Institute of TechnologyDurgapurIndia

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