Two-stage multi-item 4-dimensional transportation problem with fuzzy risk and substitution

Abstract

With the worldwide transport infrastructural development, there are several connecting roads between the cities for transportation. Some of these roads are smooth, and some are very rough, which invites some risks during transportation. For public distribution of some commodities, Govts. of developing countries collect those from the open markets, transport those to some warehouses for temporary storage, and from there, again transport to the fair-price shops for public distribution. With these facts, in this investigation, Govt.’s above scheme is modelled as two-stage four-dimensional transportation problems with different choices of conveyances and routes. There are some path and conveyance-wise uncertain travel risks. In fair-price shops, substitutable essential items (say rice and wheat) are sold with the necessary ones (say, mustard oil, etc.), which are sold by push-sale against some concession in an essential item (say, more rice, etc.). Thus, there is a trade-off between the ‘push-sale’ and ‘concession’, and hence an optimum decision is derived. Several models are developed depending on management’s and customers’ decisions, substitutability, push-sale, and environments. For road risk, a two-parameter (road’s roughness and distance) Mamdani-type fuzzy logic is implemented. The system’s profits for different models are maximised in both crisp and fuzzy environments and illustrated numerically using Generalised Reduced Gradient method through Lingo software. Here, the Govt.’s policy of introducing new products (termed as necessary item) among the public against a concession in an essential item has been introduced and analysed. This push-sale is effective up to a certain level, and after that, profit goes down.

A Preliminaries

Possibility in fuzzy environment: If R be the set of real numbers and $\tilde{P}$ and $\tilde{Q}$ be two fuzzy numbers with membership functions $\mu _{\tilde{P}}$ and $\mu _{\tilde{Q}}$, respectively. Then according to Zadeh (1978), Liu & Iwamura (1998), Dubois & Prade (2012), $Pos(\tilde{P}*\tilde{Q})$=$sup\{min(\mu _{\tilde{P}}(x)$, $\mu _{\tilde{Q}}(y))$, $x,y, \in \Re ,x*y \}$, where Pos defined possibility.

Lemma 1

Let $a=(q^1,q^2,q^3)$ be a TFN with $q^1 > 0$ and f be a crisp number, then $Pos(\tilde{a}\le f)\ge \alpha$ iff $\frac{f-q^1}{q^2-q^1}\ge \alpha$, $(q^1 \le f \le q^2)$.

Lemma 2

Let $a=(q^1,q^2,q^3)$ be a TFN with $q^1 > 0$ and f be a crisp number, then $Pos(\tilde{a}\ge f)\ge \alpha$ iff $\frac{q^3-f}{q^3-q^2}\ge \alpha$, $(q^2 \le f \le q^3)$.

Lemma 3

Let $a=(q^1,q^2,q^3)$ and $f=(f_1,f_2,f_3)$ be TFN with $q^1 > 0$ and $f_1 >0$, then $Pos(\tilde{a}\le \tilde{f})\ge \alpha$ iff $\frac{q^3-f_1}{f_2-f_1+q^3-q^2}\ge \alpha$, $(q^2<f_2,q^3>f_1)$.