A solid transportation problem in uncertain environment involving type-2 fuzzy variable

Abstract

The main focus of this paper is to develop a new safety-based restricted fixed charge solid transportation problem with type-2 fuzzy parameter that minimizes both cost and time. Here we develop mainly two models, the first one has cost and time as type-2 fuzzy variables and the second one has cost, time and all the other parameters of the solid transportation problem as type-2 fuzzy variables. We also consider restrictions on the amount of transport goods. Both of these models are solved by two different techniques. First is using the usual credibility measure, and second is the generalized credibility measure. For the first technique, we use critical value (CV)-based reduction method to reduce a type-2 fuzzy set into a type-1 fuzzy set and then apply the centroid method to this reduced fuzzy set to find the corresponding crisp value. In the second case, a chance constrained programming model based on generalized credibility has been developed with the help of CV-based reduction method. The equivalent parametric programming problem in deterministic form is then solved under the weighted mean programming technique framework, the global criteria method and with the help of LINGO 13.0 software. Lastly, we have provided two real-life-based numerical examples to illustrate the models and also validate the results with the existing work. Some sensitivity analyses for the models are also presented with some logical comments. Finally the effects of total cost and time due to the changes of credibility levels of cost, time, demand, source, conveyance and safety are discussed.

Appendix: Details on type-2 fuzzy set

A type-1 fuzzy set is a set whose elements have degrees of membership. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a MF valued in the real unit interval [0, 1]. In this whole paper the word “fuzzy” defines this fuzzy variable and denoted $\tilde{A}$.

1.1 Type-2 fuzzy set (T2 FS)

Type-2 fuzzy set $\tilde{A}$ defined on a universe of discourse $X$, which is denoted as $\tilde{A} \subseteq X,$ is a set of pairs $\left\{ {x, \mu_{{\tilde{A}}} \left( x \right)} \right\},$ where $x$ an element of a fuzzy set is, and its grade of membership $\mu_{{\tilde{A}}} \left( x \right)$ in the fuzzy set $\tilde{A}$ is a type-1 fuzzy set defined in the interval $J_{x} \subset \left[ {0, 1} \right],$ i.e., A T2 FS $\tilde{A}$ defined by Mendel and John [33] is

$$ \tilde{A} = \left\{ {\left( {\left( {x, u} \right), \mu_{{\tilde{A}}} \left( {x,u} \right)} \right): \quad \forall x \in X, J_{x} \subset \left[ {0, 1} \right]} \right\}, $$

where $0 \le \mu_{{\tilde{A}}} \left( {x,u} \right) \le 1$ is the type-2 MF.

For numerical examples on type-2 fuzzy set readers are referred to Kundu et al. [20].

1.2 Regular fuzzy variable (RFV)

For a possibility space [34] (φ, p, Pos), a regular fuzzy variable $\tilde{\xi }$ is defined as a measurable map from φ to [0, 1] in the sense that for every $t$ ∈ [0, 1], one has $\left\{ {\gamma \in \varphi | \tilde{\xi }\left( \gamma \right) \le t} \right\} \in p.$

A discrete RFV is represented as $\tilde{\xi }\sim\left( {\begin{array}{*{20}c} {r_{1} } & \ldots & {r_{n} } \\ {\mu_{1} } & \ldots & {\mu_{n} } \\ \end{array} } \right),$ where $r_{i} \in \left[ {0, 1} \right] \,{\text{and }}\, \mu_{i} > 0, \forall i\, {\text{and}}\, {\text{max}}_{i } \left\{ {\mu_{i} } \right\} = 1$.

If $\tilde{\xi } = \left( {r_{1} , r_{2} , r_{3} } \right)$ with 0 ≤ $r_{1} < r_{2} < r_{3} \le 1,$ then $\tilde{\xi }$ is called a triangular RFV.

1.3 Critical values (CVs) for RFVs

Qin et al. [30] introduced three kinds of critical values (CVs). Let $\tilde{\xi }$ be a RFV. Then:

• 1. The optimistic CV of $\tilde{\xi }$, denoted by CV*[$\tilde{\xi }$], is given by,

  

  (15)

• 2. The pessimistic CV of $\tilde{\xi }$, denoted by $CV_{*}$[$\tilde{\xi }$], is given by,

  

  (16)

• 3. The CV of $\tilde{\xi }$, denoted by CV[$\tilde{\xi }$], is given by

  

  (17)

Numerical examples of critical values are available in Kundu et al. [20].

1.4 The following theorems introduced the critical values (CVs) of trapezoidal and triangular RFVs

1.4.1 Theorem (Qin et al. [30])

Let $\tilde{\xi } = \left( {r_{1} ,r_{2} , r_{3} ,r_{4} } \right)$ be a trapezoidal RFV. Then we have

• 1. The optimistic CV of $\tilde{\xi }$ is ${\text{CV}}^{*} \left[ { \tilde{\xi }} \right] = r_{4} /\left( {1 + r_{4} - r_{3} } \right)$.

• 2. The pessimistic CV of $\tilde{\xi }$ is ${\text{CV}}_{ *}$$\tilde{\xi }$ = $r_{2} /\left( {1 + r_{2} - r_{1} } \right)$.

• 3. The CV of $\tilde{\xi }$ is

  $$ {\text{CV}}\left[ { \tilde{\xi }} \right] = \left\{ {\begin{array}{*{20}l} {\frac{{ 2r_{2} - r_{1} }}{{1 + 2\left( {r_{2} - r_{1} } \right)}},} \hfill & {{\text{if}}\quad r_{2} > \frac{1}{2} } \hfill \\ {\frac{1}{2},} \hfill & { {\text{if}}\quad r_{2} \le \frac{1}{2} < r_{3} } \hfill \\ {\frac{{r_{4} }}{{\left( {1 + 2\left( {r_{4} - r_{3} } \right)} \right)}} } \hfill & {{\text{if}}\quad r_{3} \le \frac{1}{2}} \hfill \\ \end{array} } \right. $$

For numerical examples readers are referred to Qin et al. [30].

1.4.2 Theorem (Qin et al. [30])

Let $\tilde{\xi } = \left( {r_{1} ,r_{2} , r_{3} } \right)$ be a triangular RFV. Then we have:

• 1. The optimistic CV of $\tilde{\xi }$ is ${\text{CV}}^{*} \left[ { \tilde{\xi }} \right] = r_{3} /\left( {1 + r_{3} - r_{2} } \right)$.

• 2. The pessimistic CV of $\tilde{\xi }$ is ${\text{CV}}_{ *} \left[ { \tilde{\xi }} \right] = r_{2} /\left( {1 + r_{2} - r_{1} } \right)$.

• 3. The CV of $\tilde{\xi }$ is

  $$ {\text{CV}}\left[ { \tilde{\xi }} \right] = \left\{ {\begin{array}{*{20}l} {\frac{{ 2r_{2} - r_{1} }}{{1 + 2\left( {r_{2} - r_{1} } \right)}},} \hfill & {{\text{if}}\quad r_{2} > \frac{1}{2}} \hfill \\ {\frac{{ r_{3} }}{{1 + 2\left( {r_{3} - r_{2} } \right)}},} \hfill & {{\text{if}}\quad r_{2} \le \frac{1}{2}} \hfill \\ \end{array} } \right. $$

For numerical examples readers are referred to Qin et al. [30].

1.5 CV-based reduction method for type-2 fuzzy variable

In type-2 fuzzy set, the MF itself is a fuzzy set. So computation related to type-2 fuzzy is a very difficult job. To avoid this difficulty, some defuzzification methods and methodologies have been used for defuzzification of type-2 fuzzy variable. Since we cannot apply the methodologies that are related to type-1 fuzzy sets directly to the type-2 fuzzy sets, we reduce the type-2 fuzzy sets into type-1 fuzzy sets at first and then apply the methodologies. There are several researchers who have developed different methods to defuzzify a type-2 fuzzy sets. Recently Qin et al. [30] introduced a new method named as CV-based reduction method that reduces type-2 fuzzy variables into a type-1 fuzzy variable which may or may not be normal. This method is basically based to find out three critical values and these are optimistic CV denoted as ${\text{CV}}^{*} \left[ { \tilde{\xi }} \right]$, pessimistic CV denoted as ${\text{CV}}_{ *} \left[ { \tilde{\xi }} \right]$ and CV reduction denoted as ${\text{CV}}\left[ { \tilde{\xi }} \right]$. Using these critical values we easily reduce a type-2 fuzzy variable into a type-1 fuzzy variable. The detail explanation of CV reduction method with an example is presented in Qin et al. [30].

1.5.1 Theorem (Qin et al. [30])

Let $\tilde{\xi } = \left( {r_{1} , r_{2} , r_{3} ; \theta_{l} , \theta_{r} } \right)$ be a type-2 triangular fuzzy variables. Then we have:

• 1. Using the optimistic CV reduction method, the reduction $\xi_{1}$ of $\tilde{\xi }$ has the following possibility distribution

  $$ \mu_{{\tilde{\xi }_{1} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{(1 + \theta_{r} )(x - r_{1} )}}{{r_{2} - r_{1} + \theta_{r} (x - r_{1} )}},} \hfill & {{\text{if}}\quad x \in \left[ { r_{1} ,\frac{{r_{1} + r_{2} }}{2} } \right]} \hfill \\ {\frac{{\left( {1 - \theta_{r} } \right)x + \theta_{r} r_{2} - r_{1} }}{{r_{2} - r_{1} + \theta_{r} \left( {r_{2} - r_{1} } \right)}},} \hfill & {{\text{if}}\quad x \in \left( { \frac{{r_{1} + r_{2} }}{2},r_{2} } \right]} \hfill \\ {\frac{{\left( { - 1 + \theta_{r} } \right)x - \theta_{r} r_{2} + r_{3} }}{{r_{3} - r_{2} + \theta_{r} \left( {x - r_{2} } \right)}},} \hfill & {{\text{if}}\quad x \in \left[ { r_{2} ,\frac{{r_{2} + r_{3} }}{2} } \right]} \hfill \\ {\frac{{\left( {1 + \theta_{r} } \right)( r_{3} - x)}}{{r_{3} - r_{2} + \theta_{r} \left( {r_{3} - x} \right)}},} \hfill & {{\text{if}}\quad x \in \left( { \frac{{r_{2} + r_{3} }}{2},r_{3} } \right]} \hfill \\ \end{array} } \right. $$

• 2. Using the pessimistic CV reduction method, the reduction $\xi_{2}$ of $\tilde{\xi }$ has the following possibility distribution

  $$ \mu_{{\xi_{2} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{(x - r_{1} )}}{{r_{2} - r_{1} + \theta_{l} (x - r_{1} )}},} \hfill & {{\text{if}}\quad x \in \left[ { r_{1} ,\frac{{r_{1} + r_{2} }}{2} } \right]} \hfill \\ {\frac{{(x - r_{1} )}}{{r_{2} - r_{1} + \theta_{l} \left( {r_{2} - x} \right)}},} \hfill & {{\text{if}}\quad x \in \left( { \frac{{r_{1} + r_{2} }}{2},r_{2} } \right]} \hfill \\ {\frac{{(r_{3} - x)}}{{r_{3} - r_{2} + \theta_{l} \left( {x - r_{2} } \right)}},} \hfill & {{\text{if}}\quad x \in \left[ { r_{2} ,\frac{{r_{2} + r_{3} }}{2} } \right]} \hfill \\ {\frac{{( r_{3} - x)}}{{r_{3} - r_{2} + \theta_{l} \left( {r_{3} - x} \right)}},} \hfill & {{\text{if}}\quad x \in \left( { \frac{{r_{2} + r_{3} }}{2},r_{3} } \right]} \hfill \\ \end{array} } \right. $$

Using the CV reduction method, the reduction $\xi_{3}$ of $\tilde{\xi }$ has the following possibility distribution

$$ \mu_{{\xi_{3} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{(1 + \theta_{r} )(x - r_{1} )}}{{r_{2} - r_{1} + 2\theta_{r} (x - r_{1} )}},} \hfill & {{\text{if}}\quad x \in \left[ { r_{1} ,\frac{{r_{1} + r_{2} }}{2} } \right]} \hfill \\ {\frac{{\left( {1 - \theta_{r} } \right)x + \theta_{r} r_{2} - r_{1} }}{{r_{2} - r_{1} + 2\theta_{l} \left( {r_{2} - x} \right)}},} \hfill & {{\text{if}}\quad x \in \left( { \frac{{r_{1} + r_{2} }}{2},r_{2} } \right]} \hfill \\ {\frac{{\left( { - 1 + \theta_{l} } \right)x - \theta_{l} r_{2} + r_{3} }}{{r_{3} - r_{2} + 2\theta_{l} \left( {x - r_{2} } \right)}},} \hfill & {{\text{if}}\quad x \in \left[ { r_{2} ,\frac{{r_{2} + r_{3} }}{2} } \right]} \hfill \\ {\frac{{\left( {1 + \theta_{r} } \right)( r_{3} - x)}}{{r_{3} - r_{2} + 2\theta_{r} \left( {r_{3} - x} \right)}},} \hfill & {{\text{if}}\quad x \in \left( { \frac{{r_{2} + r_{3} }}{2},r_{3} } \right]} \hfill \\ \end{array} } \right. $$

It can be noted that type-1 fuzzy variable obtained by CV-based reduction methods is not always normalized. For such cases, we cannot use the usual credibility measure; here we have to use generalized credibility measure $\widetilde{C}r$.

The following theorem finds the crisp equivalent forms of constraints involving type-2 triangular fuzzy variables, using generalized creditability measure for the reduced fuzzy variable from type-2 triangular fuzzy variable by CV reduction method.

1.5.2 Theorem (Qin et al. [30])

Suppose $\xi_{i}$ be the reduction of type-2 fuzzy variable $\tilde{\xi }_{i} = \left( {\tilde{r}_{1}^{i} ,\tilde{r}_{2}^{i} ,\tilde{r}_{3}^{i} ; \theta_{l, i} , \theta_{r, i} } \right)$ obtained by the CV reduction method for $i = 1, 2, . . . , n$ and $\xi_{1} , \xi_{2} , . . . . , \xi_{n}$ are mutually independent, and $k_{i} \ge 0$ for $i = 1, 2, . . . , n$.

• 1. Given the generalized credibility level $\alpha \in \left( {0, 0.5} \right]$, if $\alpha \in \left( {0, 0.25} \right]$, then $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ is equivalent to

  $$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 - 2\alpha + \left( {1 - 4\alpha } \right)\theta_{r, i} } \right)k_{i} r_{1}^{i} + 2\alpha k_{i} r_{2}^{i} }}{{1 + \left( {1 - 4\alpha } \right)\theta_{r, i} }} \le t, $$

  and if $\alpha \in \left( {0.25, 0.5} \right]$, then $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ is equivalent to

  $$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 - 2\alpha } \right)k_{i} r_{1}^{i} + \left( {2\alpha + \left( {4\alpha - 1} \right)\theta_{l, i} } \right)k_{i} r_{2}^{i} }}{{1 + \left( {4\alpha - 1} \right)\theta_{l, i} }} \le t, $$

• 2. Given the generalized credibility level $\alpha \in \left( {0.5, 1} \right]$, if $\alpha \in \left( {0.5, 0.75} \right]$, then $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ is equivalent to

  $$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {2\alpha - 1} \right)k_{i} r_{3}^{i} + \left( {2\left( {1 - \alpha } \right) + \left( {3 - 4\alpha } \right)\theta_{l, i} } \right)k_{i} r_{2}^{i} }}{{1 + \left( {3 - 4\alpha } \right)\theta_{l, i} }} \le t, $$

  and if $\alpha \in \left( {0.75, 1} \right]$, then $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ is equivalent to

  $$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {2\alpha - 1 + \left( {4\alpha - 3} \right)\theta_{r, i} } \right)k_{i} r_{3}^{i} + 2\left( {1 - \alpha } \right)k_{i} r_{2}^{i} }}{{1 + \left( {4\alpha - 3} \right)\theta_{r, i} }} \le t. $$

1.5.2.1 Corollary

With the help of the above theorem, we can also find an equivalent expression for $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ as follows:

As we know,

$$ \widetilde{Cr}\left\{ {\mathop \sum \limits_{i = 1}^{n} k_{i} \xi_{i} \ge t} \right\} \ge \alpha \Rightarrow \widetilde{Cr}\left\{ {\mathop \sum \limits_{i = 1}^{n} - k_{i} \xi_{i} \le - t} \right\} \ge \alpha \Rightarrow \widetilde{Cr}\left\{ {\mathop \sum \limits_{i = 1}^{n} k_{i} \xi_{i}^{\prime } \le t^{\prime } } \right\} \ge \alpha , $$

where $\xi_{i}^{\prime } = - \xi_{i}$ is the reduction of $- \, \tilde{\xi }_{i} = \left( { - \,\tilde{r}_{1}^{i} , - \,\tilde{r}_{2}^{i} , - \,\tilde{r}_{3}^{i} ; \theta_{r, i} , \theta_{l, i} } \right)$ and $- \,t = t^{\prime }$.

Now using [21] of the theorem (“Regular fuzzy variable (RFV)” section in Appendix ), given the generalized credibility level $\alpha \in \left( {0, 0.5} \right]$, if $\alpha \in \left( {0, 0.25} \right]$, then $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ is equivalent to

$$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 - 2\alpha + \left( {1 - 4\alpha } \right)\theta_{l, i} } \right)k_{i} \left( { - r_{3}^{i} } \right) + 2\alpha k_{i} \left( { - r_{2}^{i} } \right)}}{{1 + \left( {1 - 4\alpha } \right)\theta_{l, i} }} \le t^{\prime } = - t, $$

which implies

$$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 - 2\alpha + \left( {1 - 4\alpha } \right)\theta_{l, i} } \right)k_{i} \left( {r_{3}^{i} } \right) + 2\alpha k_{i} \left( {r_{2}^{i} } \right)}}{{1 + \left( {1 - 4\alpha } \right)\theta_{l, i} }} \ge t, $$

and if $\alpha \in \left( {0.25, 0.5} \right]$, then $\widetilde{C}r\left\{ {\sum\nolimits_{i = 1}^{n} {k_{i} \xi_{i} \le t} } \right\} \ge \alpha$ is equivalent to

$$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 - 2\alpha } \right)k_{i} \left( { - r_{3}^{i} } \right) + \left( {2\alpha + \left( {4\alpha - 1} \right)\theta_{r, i} } \right)k_{i} \left( { - r_{2}^{i} } \right)}}{{1 + \left( {4\alpha - 1} \right)\theta_{r, i} }} \le - t, $$

which implies

$$ \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 - 2\alpha } \right)k_{i} \left( {r_{3}^{i} } \right) + \left( {2\alpha + \left( {4\alpha - 1} \right)\theta_{r, i} } \right)k_{i} \left( {r_{2}^{i} } \right)}}{{1 + \left( {4\alpha - 1} \right)\theta_{r, i} }} \le t, $$

For different values of $\alpha$, the similar equivalent expression can be obtained.

1.6 Defuzzification of a type-2 fuzzy variable by CV-based reduction method

The defuzzification process of a type-2 fuzzy variable has two stages. In the first stage, the type-2 fuzzy variable is reduced to its corresponding type-1 fuzzy variable and in the second stage the crisp value is obtained by applying different defuzzification methods like as centroid method [35], expected value method [36, 37] to the reduced fuzzy variables. In this paper, we first apply the CV-based reduction method to the type-2 fuzzy variables, so that we get type-reduced form, i.e., a type-1 fuzzy variable and then we apply the centroid method to the type-1 fuzzy variables, resulting a crisp value.

1.6.1 Centroid defuzzification technique

The centroid method is also known as center of gravity or center of area defuzzification. It was first proposed by Sugeno [38] in 1985. It is the most commonly used method and is more accurate compared to other existing methods. The method can be expressed as

$$ x^{*} = \left\{ {\begin{array}{*{20}l} {\frac{{\mathop \sum \nolimits_{i = 1}^{n} x_{i} \mu_{\xi } \left( {x_{i} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \mu_{\xi } \left( {x_{i} } \right)}},} \hfill & {{\text{for}}\,{\text{discrete}}\,{\text{case}},} \hfill \\ {\frac{{\smallint x_{i} \mu_{\xi } \left( {x_{i} } \right){\text{d}}x}}{{\smallint \mu_{\xi } \left( {x_{i} } \right)}}, } \hfill & {{\text{for}}\,{\text{continuous}}\, {\text{case}}.} \hfill \\ \end{array} } \right. $$

where $\xi = (\xi_{1} , \xi_{2} , \ldots , \xi_{n} )$ is a fuzzy variable, $x^{*}$ is the corresponding crisp value to be obtained, $\mu_{\xi } \left( {x_{i} } \right)$ is the aggregated MF, and x is the output variable.

Numerical examples of crisp conversion of TRS using the centroid method are available in Kundu et al. [20].

Here it should be mentioned that the ${\text{CV}}$-based reduction method gives the more centroid compromised crisp value, compared to the optimistic ${\text{CV}}^{*}$ and pessimistic ${\text{CV}}_{*}$ as these values are evaluated using the possibility and necessity measures, respectively, whereas the ${\text{CV}}$ reduction method is based on the average of these two measures.

The entire defuzzification process of a type-2 fuzzy set is depicted in Fig. 4.

Fig. 4

Defuzzification process of a type-2 fuzzy variable