OPSEARCH

, Volume 56, Issue 2, pp 583–602

# Total cost measures with probabilistic cost function under varying supply and demand in transportation problem

Application Article

## Abstract

In the present competitive world, it is often said that “Time is Money” in almost every aspect of life. Time is a factor which affects the various real-life problems directly or indirectly. So, in order to incorporate the “time” as a factor in transportation problems (TPs), we have considered the probabilistic cost/profit function termed as “survival cost/profit” which is again a time-dependent function. In this study, we have assumed that the supply and demand quantities are varying between some specified intervals. Due to the variation in the supply and demand quantities, the value of the objective function is also obtained between interval which is bounded by lower and upper values. Based on the above-stated assumptions, we have developed a couple of mathematical optimization models for the TPs. The solution procedure has also been discussed to solve the proposed mathematical models. At last, a numerical illustration has been presented to show the validity of the model and solution procedure which is helpful in the decision-making process.

## Keywords

Probabilistic cost/profit function Varying supply and demand Transportation problem

## List of symbols

$$x_{ij}$$

Number of items transported from ith source to jth destination

$$c_{ij}$$

Transportation cost of unit item transported from ith source to jth destination

$$p_{ij}$$

Profit incurred over unit item transported from ith source to jth destination

$$S_{ij} (t)$$

Probability that items will remain in good condition when transported from ith source to jth destination which is function of time (t)

$$S_{ij} (c_{ij} )$$

Survival cost function defined from ith source to jth destination

$$S_{ij} (p_{ij} )$$

Survival profit function defined from ith source to jth destination

$$\tilde{a}_{i}$$

Varying supply quantity at ith source

$$\tilde{b}_{j}$$

Varying demand quantity at jth destination

$$[\underline{{a_{i} }} ,\overline{{a_{i} }} ]$$

Lower and upper bound on the supply quantity at ith source

$$[\underline{{b_{j} }} ,\overline{{b_{j} }} ]$$

Lower and upper bound on the demand quantity at jth destination

$$u_{i}$$

Dual variable associated with ith supply constraint

$$v_{j}$$

Dual variable associated with jth demand constraint

## Notes

### Acknowledgements

Authors are very thankful and overwhelmed to all the anonymous reviewers and editors for the insightful comments to enhance the readability of the manuscript. The first author is also very thankful to Mr. Abdul Nasir Khan for his valuable suggestions and eternal support.

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