Transportation Problem with Fuzzy Unit Costs. Z-fuzzy Numbers Approach

Abstract

In the classical version of the transportation problem it is assumed that all the parameters (the unit transport costs, capacities and amounts demanded) are deterministic. In real life problems, these parameters are not deterministic. One way of modeling such problems is by using the concept of fuzzy numbers. An approach using classical fuzzy numbers has been used in a numerous of articles. In proposed in this work transportation models, we assume that demand and supply are deterministic numbers and the uncertainty associated with the transport costs is modeled using Z-fuzzy numbers. A $Z$-fuzzy number is an ordered pair of fuzzy numbers $Z=(A,B)$. A $Z$-fuzzy number is associated with a real-valued uncertain variable, $X$, with the first component, $A$, playing the role of a fuzzy restriction, $R(X$), on the values which $X$ can take, written as $X$ is $A$, where $A$ is a fuzzy set. $B$ is a measure of reliability (certainty) of the $A$. This allows to include in the model, next to the estimated value of the parameter (here: unit transport cost value), the expert's opinion as to the expert's certainty regarding this estimation. An illustrative example is presented.