# Uncertain programming models for fixed charge multi-item solid transportation problem

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## Abstract

This paper investigates the fixed charge multi-item solid transportation problem, in which the fixed charges, direct costs, transportation capacities, supply and demand are uncertain variables. Based on the uncertainty theory, expected value programming model and chance-constrained programming model for fixed charge multi-item solid transportation problem are constructed, respectively. We can obtain the optimal solution of two models via solving the relevant deterministic models. Finally, a numerical experiment is implemented to illustrate the application of the models.

## Keywords

Transportation problem Uncertainty programming Uncertain variable## Notes

### Acknowledgements

This work was supported by the Natural Science Foundation of China (No. 11626234), the Hubei Provincial Natural Science Foundation (No. 2016CFB308), the Higher Educational Science and Technology Program Foundation of Shandong Province (No. J13LI10), and the Foundation of Liaocheng University (No. 318011303).

### Compliance with ethical standards

### Conflicts of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

## References

- Bhatia H, Swarup K, Puri M (1976) Time minimizing solid transportation problem. Math Oper Stat 7:395–403MathSciNetMATHGoogle Scholar
- Chen X, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81MathSciNetCrossRefMATHGoogle Scholar
- Chen X, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604CrossRefMATHGoogle Scholar
- Chen B, Liu Y, Zhou T (2017) An entropy based solid transportation problem in uncertain environment. J Ambient Intell Humaniz Comput. doi: 10.1007/s12652-017-0535-z
- Cui Q, Sheng Y (2012) Uncertain programming model for solid transportation problem. Information 15(12):342–348Google Scholar
- Dalman H (2017) Uncertain programming model for multi-item solid transportation problem. Int J Mach Leant Cybern. doi: 10.1007/s13042-016-0538-7
- Gao R (2016) Milne method for solving uncertain differential equations. Appl Math Comput 11(274):774–785MathSciNetGoogle Scholar
- Gao J, Yao K (2015) Some concepts and theorems of uncertain random process. Int J Intell Syst 30(1):52–65CrossRefGoogle Scholar
- Guo C, Gao J (2017) Optimal dealer pricing under transaction uncertainty. J Intell Manuf 28(3):657–665CrossRefGoogle Scholar
- Gao X, Gao Y, Ralescu D (2010) On Liu’s inference rule for uncertain systems. Int J Uncertain Fuzz 18(1):1–11MathSciNetCrossRefMATHGoogle Scholar
- Gao R, Sun Y, Ralescu D (2016) Order statistics of uncertain random variables with application to k-out-of-n system. Fuzzy Optim Decis Mak 16(2):159–181MathSciNetCrossRefGoogle Scholar
- Gao Y, Yang L, Li S (2016) Uncertain models on railway transportation planning problem. Appl Math Model 40:4921–4934MathSciNetCrossRefGoogle Scholar
- Gao J, Yang X, Liu D (2017) Uncertain Shapley value of coalitional game with application to supply chain alliance. Appl Soft Comput 56:551–556CrossRefGoogle Scholar
- Haley K (1962) The solid transportation problem. Oper Res 11:446–448MATHGoogle Scholar
- Hirsch W, Dantzig G (1968) The fixed charge problem. Naval Res Logist Quart 15:413–424MathSciNetCrossRefMATHGoogle Scholar
- Kennington J, Unger V (1976) A new branch and bound algorithm for the fixed charge transportation problem. Manag Sci 22:1116–1126MathSciNetCrossRefMATHGoogle Scholar
- Li X, Qin Z (2014) Interval portfolio selection models within the framework of uncertainty theory. Econ Model 41:338–344CrossRefGoogle Scholar
- Li Y, Ida K, Gen M, Kobuchi R (1997) Neural network approach for multicriteria solid transportation problem. Comput Ind Eng 33(3–4):465–468CrossRefGoogle Scholar
- Liu B (2007) Uncertainty theory, 2nd edn. Springer, BerlinMATHGoogle Scholar
- Liu B (2008) Fuzzy process: hybrid process and uncertain process. J Uncertain Syst 2(1):3–16Google Scholar
- Liu B (2009) Theory and practice of uncertain programming, 2nd edn. Springer, BerlinCrossRefMATHGoogle Scholar
- Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
- Liu B (2010) Uncertain set theory and uncertain inference rule with application to uncertain control. J Uncertain Syst 4(2):83–98Google Scholar
- Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinCrossRefGoogle Scholar
- Lotif MM, Moghaddam RT (2013) A genetic algorithm using priority-based encoding with new operators for fixed charge transportation problems. Appl Soft Comput 13:2711–2722CrossRefGoogle Scholar
- Ojha A, Das B, Mondal S, Maiti M (2010) A solid transportation problem for an item with fixed charge vechicle cost and price discounted varying charge using genetic algorithm. Appl Soft Comput 10:100–110CrossRefGoogle Scholar
- Romeijna HE, Sargutb FZ (2011) The stochastic transportation problem with single sourcing. Eur J Oper Res 214(2):262–272MathSciNetCrossRefGoogle Scholar
- Sheng Y, Yao K (2012) Fixed charge transportation problem and its uncertain programming model. Ind Eng Manag Syst 11(2):183–187Google Scholar
- Sheng Y, Gao Y (2016) Shortest path problem of uncertain random network. Comput and Ind Eng 99:97–105CrossRefGoogle Scholar
- Sun M, Aronson JE, McKeown PG, Drinka D (1998) A tabu search heuristic procedure for the fixed charge transportation problem. Eur J Oper Res 106:441–456CrossRefMATHGoogle Scholar
- Wang X, Ning Y, Moughal T, Chen X (2015) Adams–Simpson method for solving uncertain differential equations. Appl Math Comput 271:209–219MathSciNetGoogle Scholar
- Williams A (1963) A stochastic transportation problem. Oper Res 11:759–770Google Scholar
- Yang L, Feng Y (2007) A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl Math Model 31:2668–2683CrossRefMATHGoogle Scholar
- Yang X, Gao J (2013) Uncertain differential games with applicatians to capitalism. J Uncertain Anal Appl l: Article 17Google Scholar
- Yang X, Ralescu D (2015) Adams method for solving uncertain differential equations. Appl Math Comput 270:993–1003MathSciNetGoogle Scholar
- Yang X, Shen Y (2015) Runge-Kutta method for solving uncertain differential equations. J Uncertain Anal Appl 3 (Article 17)Google Scholar
- Yang X, Gao J (2016) Linear-quadratic uncertain differential games with applicatians to resource extraction problem. IEEE T Fuzzy Syst 24(4):819–826CrossRefGoogle Scholar
- Yang X, Gao J (2017) Bayesian equilibria for uncertain bimatrix game with asymmetric information. J Intell Manuf 28(3):515–525CrossRefGoogle Scholar
- Yao K (2013) Extreme values and integral of solution of uncertain differential equation. J Uncertain Anal Appl 1: Article 2Google Scholar
- Yao K (2015) Inclusion relationship of uncertain sets. J Uncertain Anal Appl 3: Article 13Google Scholar
- Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832MathSciNetMATHGoogle Scholar
- Zhang B, Peng J, Li S, Chen L (2016) Fixed charge solid transportation Pproblem in uncertain environment and its algorithm. Comput Ind Eng 102(2016):186–197CrossRefGoogle Scholar