# Uncertain interval programming model for multi-objective multi-item fixed charge solid transportation problem with budget constraint and safety measure

## Abstract

This paper presents uncertain interval programming models for multi-objective multi-item fixed charge solid transportation problem with budget constraint and safety measure (MOMIFCSTPBCSM). The human languages usually involve imperfect or unknown information and are in the lack of certainty, and often, it is impossible to exactly describe an existing state or a future outcome. In using the probability theory, we must have enough historical information to estimate the probability distributions and in the case of fuzzy theory, we must have a trustworthy membership function, which is not easy to do. Thus, we often estimate the degree of belief with some hesitation that each condition may occur. To deal with such a situation, the uncertain interval theory may be very useful. Based on these facts, the parameters of the formulated problem are chosen as uncertain intervals. We consider unit transportation costs, fixed charges, transportation times, deterioration of items, supplies at origins, demands at destinations, conveyance capacities, budget at each destination, selling prices and purchasing costs, and we assume the safety factor and the desired safety measure are interval uncertain parameters. To formulate the proposed MOMIFCSTPBCSM, we use interval theory and uncertain programming techniques to develop two different models: an Expected Value Model and a Chance-Constrained Model. The equivalent deterministic models are formulated and solved using a linear weighted method, a fuzzy programming method and the goal programming method.

## Keywords

Multi-objective multi-item fixed charge solid transportation problem Budget constraint Safety measure Deterioration of item Theories of interval## Notes

### Acknowledgements

The authors would like to thank the referees for their valuable comments which helped to improve the manuscript.

### 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 contain any study 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.

## References

- Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, New YorkzbMATHGoogle Scholar
- Atanassov K (1986) Intuitionistic fuzzy set. Fuzzy Sets Syst 20:87–96zbMATHGoogle Scholar
- Baidya A, Bera UK (2014) An interval valued solid transportation problem with budget constraint in different interval approaches. J Transp Secur 7(2):147–155Google Scholar
- Baidya A, Bera UK, Maiti M (2013) Solution of multi-item interval valued solid transportation problem with safety measure using different methods. Oper Res 51(1):1–22MathSciNetzbMATHGoogle Scholar
- Bhatia HL, Swarup K, Puri MC (1976) Time minimizing solid transportation problem. Mathematische operations forschung und statistik 7(3):395–403MathSciNetzbMATHGoogle Scholar
- Bit AK, Biswal MP, Alam SS (1993) Fuzzy programming approach to multi-objective solid transportation problem. Fuzzy Sets Syst 57(2):183–194zbMATHGoogle Scholar
- Cerulli R, D’Ambrosio C, Gentili M (2017) Best and worst values of the optimal cost of the interval transportation problem. Proc Math Stat 217:367–374MathSciNetGoogle Scholar
- Chakraborty D, Jana DK, Roy TK (2014) Multi-objective multi-item solid transportation problem with fuzzy inequality constraints. J Inequal Appl 338(1):1–22MathSciNetzbMATHGoogle Scholar
- Charnes A, Cooper W (1961) Management models and industrial applications of linear programming. Wiley, New YorkzbMATHGoogle Scholar
- Chen L, Peng J, Zhang B (2017) Uncertain goal programming models for bicriteria solid transportation problem. Appl Soft Comput 51:49–59Google Scholar
- Dalman H (2016) Uncertain programming model for multi-item solid transportation problem. Int J Mach Learn Cybernet 9(4):559–567MathSciNetGoogle Scholar
- Dalman H, Sivri M (2017) Multi-objective solid transportation problem in uncertain environment. Iran J Sci Technol Trans Sci 41(2):505–514zbMATHGoogle Scholar
- Dalman H, Güzel N, Sivri M (2016) A fuzzy set-based approach to multi-objective multi-item solid transportation problem under uncertainty. Int J Fuzzy Syst 18(4):716–729MathSciNetGoogle Scholar
- D’Ambrosio C, Gentili M, Cerulli R (2019) The optimal value range problem for the Interval (immune) transportation Problem. Omega. https://doi.org/10.1016/j.omega.2019.04.002
- Ebrahimnejad A (2016) Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sadhana 41(3):299–316MathSciNetzbMATHGoogle Scholar
- Gao Y, Kar S (2017) A solid transportation model with product blending. Int J Fuzzy Syst 19(6):1916–1926MathSciNetGoogle Scholar
- Hirsch WM, Dantzig GB (1968) The fixed charge problem. Naval Res Logist 15(3):413–424MathSciNetzbMATHGoogle Scholar
- Hu BQ, Wang S (2006) A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers. J Ind Manag Optim 2(4):351–371MathSciNetzbMATHGoogle Scholar
- Jiménez F, Verdegay J (1999) An evolutionary algorithm for interval solid transportation problems. Evol Comput 7(1):103–107Google Scholar
- Kar MB, Kundu P, Kar S, Pal T (2018) A multi-objective multi-item solid transportation problem with vehicle cost, volume and weight capacity under fuzzy environment. J Intell Fuzzy Syst 35(2):1991–1999Google Scholar
- Kundu P, Kar S, Maiti M (2013a) Multi-objective multi-item solid transportation problem in fuzzy environment. Appl Math Model 37(4):2028–2038MathSciNetzbMATHGoogle Scholar
- Kundu P, Kar S, Maiti M (2013b) Multi-objective solid transportation problems with budget constraint in uncertain environment. Int J Syst Sci 45(8):1668–1682MathSciNetzbMATHGoogle Scholar
- Kundu P, Kar S, Maiti M (2014) Fixed charge transportation problem with type-2 fuzzy variables. Inf Sci 255:170–186MathSciNetzbMATHGoogle Scholar
- Kundu P, Kar S, Maiti M (2015) Multi-item solid transportation problem with type-2 fuzzy parameters. Appl Soft Comput 31:61–80Google Scholar
- Kundu P, Kar MB, Kar S, Pal T, Maiti M (2017) A solid transportation model with product blending and parameters as rough variables. Soft Comput 21(9):2297–2306zbMATHGoogle Scholar
- Liu B (2007) Uncertainty theory, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
- Liu B (2009a) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
- Liu B (2009b) Theory and practice of uncertain programming, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
- Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinGoogle Scholar
- Majumder S, Kundu P, Kar S, Pal T (2018) Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint. Soft Comput 23(10):3279–3301zbMATHGoogle Scholar
- Midya S, Roy K (2017) Analysis of interval programming in different environments and its application to fixed charge transportation problem. Discrete Math Algorithms Appl 9(3):1–17MathSciNetzbMATHGoogle Scholar
- Moore RE (1979) Methods and applications of interval analysis. SIAM, PhiladephiazbMATHGoogle Scholar
- Nagarajan A, Jeyaraman K (2010) Solution of chance constrained programming problem for multi-objective interval solid transportation problem under stochastic environment using fuzzy approach. Int J Comput Appl 10(9):19–29Google Scholar
- Nagarajan A, Jeyaraman K (2014) Multi-objective solid transportation problem with interval cost in source and demand parameters. Int J Comput Organ Trends 8(1):33–41Google Scholar
- Pawlak Z (1982) Rough sets. Int J Inf Comput Sci 11(5):341–356zbMATHGoogle Scholar
- Roy K, Midya S (2019) Multi-objective fixed-charge solid transportation problem with product blending under intuitionistic fuzzy environment. Appl Intell 49(10):3524–3538Google Scholar
- Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353zbMATHGoogle Scholar
- Zadeh LA (1975a) The concept of a linguistic variable and its application to approximate reasoning–I. Inf Sci 8(3):199–249MathSciNetzbMATHGoogle Scholar
- Zadeh LA (1975b) The concept of a linguistic variable and its application to approximate reasoning–II. Inf Sci 8(4):301–357MathSciNetzbMATHGoogle Scholar