The Inventory Pollution-Routing Problem Under Uncertainty

Abstract

Carbon emissions from supply chain operations are extensively contributing to the global warming. Sustainable supply chain management literature has seen more emphasis on greening of production operations and designing of greener supply networks, considering transportation emissions as “necessary evil”. This chapter aims to investigate the economic and environmental consequences of transport routing decisions in a supply chain with vertical collaboration, for instance through Vendor Managed Inventory. An optimization model and solution method is presented for an Inventory Pollution-Routing Problem (IPRP) in which inventory and transportation costs and emissions as well as demand uncertainty concerns are explicitly incorporated. The proposed model can be used to explore possible tradeoffs between emissions costs and operational costs for green inventory routing decision making. A set of computational tests are designed for performance benchmark of the proposed model and solution method.

Appendix

1.1 Fuzzy Number

The theory of fuzzy sets introduced by Zadeh (1965) was developed to describe vagueness and ambiguity in the real world system. Zadeh defined a fuzzy set ${{\tilde{a}}}$ in a universe of discourse X as a class of objects with a continuum of grades of memberships. Such a set is characterized by a membership function $\mu_{{\tilde{a}}} (x)$ which associates with each point x in X a real number in the interval [0,1]. $\mu_{{\tilde{a}}} (x)$ represents the grade of membership of x in ${{\tilde{a}}}.$ A fuzzy set ${{\tilde{a}}}$ in the universe of discourse R (set of real numbers) is called a fuzzy number if it satisfies the following conditions:

• (i) ${{\tilde{a}}}$ is normal i.e. there exists at least one $x \in {\text{R}}$ such that $\mu_{{\tilde{a}}} (x) = 1.$

• (ii) ${{\tilde{a}}}$ is convex.

• (iii) the membership function $\mu_{{{\tilde{a}}}} (x) , x \in {\text{R}}$ is at least piecewise continuous.

1.1.1 Triangular Fuzzy Number

Triangular fuzzy number (TFN) $({{\tilde{a}}})$ is the fuzzy number with the membership function $\mu_{{{\tilde{a}}}} (x) ,$ a continuous mapping: $\mu_{{{\tilde{a}}}} (x) : {\text{R}} \to [0,1] ,$ where

$$\mu_{{\tilde{a}}} (x) = \left\{ {\begin{array}{*{20}l} 0 \hfill & \{ - \infty < x < a_{1} \} \hfill \\ {\frac{{x - a_{1} }}{{a_{2} - a_{1} }}} \hfill & {a_{1} \le x < a_{2} } \hfill \\ {\frac{{a_{3} - x}}{{a_{3} - a_{2} }}} \hfill & {a_{2} \le x \le a_{3} } \hfill \\ 0 \hfill & {a_{3} < x < \infty .} \hfill \\ \end{array} } \right.$$

(6.38)

1.1.2 α-Cut of a Fuzzy Number

An α-cut of a fuzzy number ${{\tilde{a}}}$ is defined as a crisp set

$$a_{\alpha } = \left\{ {x{:}\, \mu_{{\tilde{a}}} (x) \ge \alpha ,\quad x \in {\text{R}}} \right\}\quad {\text{where}}\,\eta \;\left[ {0,1} \right].$$

1.1.3 Approximate Value of Triangular Fuzzy Number (TFN)

According to Kaufmann and Gupta (1991), the approximated value of TFN $\tilde{a} \equiv \left( {a_{1} ,a_{2} ,a_{3} } \right)$ is given by $\hat{a} = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}\left( {a_{1} + 2a_{2} + a_{3} } \right).$

1.1.4 Algebraic Operation of Fuzzy Numbers

Addition

Let $\tilde{a} \equiv \left( {a_{1} ,a_{2} ,a_{3} } \right)$ and $\underline{{\tilde{b}}} \equiv \left( {b_{1} ,b_{2} ,b_{3} } \right)$ be two triangular fuzzy numbers. Using max-min convolution on fuzzy numbers ${{\tilde{a}}}$ and ${{\tilde{b}}}$ the membership function of the resulting fuzzy number ${{\tilde{a}}} \, ({+}) \, {{\tilde{b}}}$ can be obtained as $\vee_{z = x + y} \left( {\mu_{{\tilde{a}}} (x) \wedge \mu_{{\tilde{b}}} (y)} \right),\;\forall_{x,y,z} \in {\text{R}}$ where the symbols ‘$\wedge$’ and ‘$\vee$’ are used for minimum and maximum, respectively. In short we can write $\tilde{a} \, ( + ) \, \tilde{b} = \left( {a_{1} ,a_{2} ,a_{3} } \right) ( { + )}\left( {b_{1} ,b_{2} ,b_{3} } \right).$

Scalar multiplication

For any real constant t,

$$t\tilde{a} = \left\{ \begin{aligned} (ta_{1} ,ta_{2} ,ta_{3} )\quad t \ge 0 \hfill \\ (ta_{3} ,ta_{2} ,ta_{1} )\quad t < 0. \hfill \\ \end{aligned} \right.$$

1.1.5 Fuzzy Possibility Techniques

Let ${{\tilde{a}}}$ and ${{\tilde{b}}}$ be two fuzzy quantities with membership functions $\mu_{{\tilde{a}}} (x)$ and $\mu_{{\tilde{b}}} (y),$ respectively. Then according to Dubois and Prade (1980), Liu and Iwamura (1998a, b) $pos\left( {\tilde{a}*\tilde{b}} \right) = \sup \left\{ {\hbox{min} \left( {\mu_{{\tilde{a}}} (x),\mu_{{\tilde{b}}} (y)} \right) {:}\, x,y \in {\text{R}}, \, x*y} \right\},$ where the abbreviation ‘pos’ represents possibility and * is any of the relations <, >, =, ≤, ≥.

If ${\tilde{a}}$ and ${\tilde{b}}$ are two fuzzy numbers defined on R and ${\tilde{u}} = f({\tilde{a}}, {\tilde{b}})$ where $f {:} \, {\text{R}} \times {\text{R}} \to {\text{R}}$ is a binary operation then the membership function $\mu_{\tilde{u}}$ of ${\tilde{u}}$ is defined as $\mu_{\tilde{u}} (u) = \sup \{ {\min(\mu_{\tilde{a}}(x), \mu_{\tilde{b}}(y)) {:}\, x, y \in {\text{R}} }$ and u = f(x,y), ∀u ∈ R}.

1.2 Random Variable

Let $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} \, ( {=} (m,\sigma^{2} ))$ be a continuous random variable with probability density function (PDF) $f_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }} \left( l \right)$ whose mean and variance are m and σ ², respectively. Similarly, let $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}^{\prime} \,( {=} (m^{\prime} , \sigma^{\prime 2} ))$ be another random variable with pdf $f_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L^{\prime}} }} (l^{\prime}).$ If ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }$ and ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L^{\prime}} }$ are two independent random variables, then we have the following algebraic operations:

• Addition:

$$\underline{L}_{1} [ + ]\underline{L}_{2} = (m_{1} ,\sigma_{1}^{2} )[ + ]{\kern 1pt} {\kern 1pt} (m_{2} ,\sigma_{2}^{2} ) = \,(m_{1} + m_{2} ,\sigma_{1}^{2} + \sigma_{2}^{2} ).$$

Here, according to sum-product convolution $\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }( = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}^{\prime})$ is a random variable with the same type of pdf $f_{{\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }}} \left( \varvec{l} \right) = (\int\nolimits_{\text{R}} f(\varvec{l} - l^{\prime})f^{\prime} (l^{\prime})dl^{\prime}$ with mean $\varvec{m^{\prime}}^{2} ( {=} m^{2} + m^{\prime} )$ and variance $\varvec{\sigma}^{\prime 2} ( = \sigma^{2} + \sigma^{\prime 2} ) .$

• Scalar multiplication:

$t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} = (tm,t^{2} \sigma^{2} ).$ Here tL and L have the same type of PDF.

1.3 Hybrid Number (Kaufmann and Gupta 1991)

Assume $\tilde{A}( = (\tilde{A},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} ))$ is a hybrid number. Here the couple ($\tilde{A},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}$) represents the addition to a fuzzy number with a random variable without altering the characteristic of each one and without decreasing the amount of available information where à is a fuzzy number and L is the random variable with density function $f_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }} \left( l \right).$ Let $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tilde{A}} ( = (\tilde{A},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} ))$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tilde{A}}^{\prime}( = (\tilde{A}^{\prime},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}^{\prime}))$ be two hybrid numbers in R where $f_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }} \left( l \right)$ and $f_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}^{\prime}}} (l^{\prime})$ are the pdfs of L and L′, respectively. So a hybrid convolution for addition will be defined as $(\tilde{A},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} ) \oplus (\tilde{A}{\prime} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}^{\prime}) = (\tilde{A}\left( + \right)\tilde{A}^{\prime} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} [ + ]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}^{\prime}) = (\tilde{\varvec{A}},\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} }),$ where (+) represents the max-min convolution for addition of fuzzy subsets and [+] represents the sum-product convolution for addition of random variables. We denote the couple $(\tilde{\varvec{A}},\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} })$ by the symbol $\tilde{\varvec{A}}( + )^{\prime}\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} } .$

So,${\kern 1pt} \mu_{{\tilde{A}_{1} ( + )\tilde{A}_{2} }} (z) = \vee_{z = x + y} (\mu_{{\tilde{A}_{1} }} (x) \wedge \mu_{{\tilde{A}_{2} }} (y)),\forall x,y,z \in \text{R}$ and $f(l) = \int\nolimits_{\text{R}} f_{1} (l - l_{2} )f_{2} (l_{2} )dl_{2}$ or $\int\nolimits_{\text{R}} f_{1} (l_{1} )f_{2} (l - l_{1} )dl_{1} .$

Note 1

A fuzzy number is a special case of a hybrid number if $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\tilde{A}} = (\tilde{A},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} ),$ where 0 is the trivial random variable with the following probabilities:

$$\begin{aligned} P(l) & = 1,\quad l = 0 \\ & = 0{\kern 1pt} ,\quad l \ne 0. \\ \end{aligned}$$

Note 2

A random variable is also a special case of a hybrid number if $\tilde{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} } = \left( {\tilde{0},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} } \right),$ where ${\tilde{0}}$ is the trivial fuzzy number with membership function

$$\begin{aligned} \mu_{{\tilde{0}}} (x) & = 1,\quad x = 0 \\ & = 0{\kern 1pt} ,\quad x \ne 0. \\ \end{aligned}$$

Note 3

$\underline{{\tilde{0}}} = (0, 0)$ is the neutral for addition of hybrid numbers.

If $\tilde{u}_{1}$ is a fuzzy cost, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}_{2}$ is a random cost and u ₃ is a fixed cost then the total cost can be expressed as

$$\tilde{u}_{1} [ + ]{\kern 1pt} \underline{u}_{2} [ + ]{\kern 1pt} u_{3} = (\tilde{u}_{1} ,0){\kern 1pt} [ + ]{\kern 1pt} (0,\underline{u}_{2} ){\kern 1pt} [ + ]{\kern 1pt} (0,u_{3} ) = (\tilde{u}_{1} ,\underline{u}_{2} ( + )^{\prime}u_{3} ) = (\tilde{u}_{1} ( + )u_{3} ,\underline{u}_{2} ){\kern 1pt} .$$

(6.39)

We can consider the fixed number like a sum of two parts $u_{3} = u^{\prime}_{3} + u^{\prime\prime}_{3}$ and write for (6.39)

$$\tilde{u}_{1} [ + ]{\kern 1pt} \underline{u}_{2} [ + ]{\kern 1pt} u_{3} = (\tilde{u}_{1} [ + ]{\kern 1pt} u^{\prime}_{3} ,\underline{u}_{2} [ + ]{\kern 1pt} u^{\prime\prime}_{3} ).$$

(6.40)

The mathematical expectation of a hybrid number is defined as follows.

A function ϕ(x) in R that is nonnegative and monotonically increasing is:

$$\begin{aligned} & \forall x_{1} ,x_{2} \in {\text{R}}: \\ & (x_{1} > x_{2} ) \Rightarrow (\varphi (x_{2} ) \ge \varphi (x_{1} )){\kern 1pt} . \\ \end{aligned}$$

(6.41)

For a closed interval of R, $[a_{\alpha }^{1} ,a_{\alpha }^{2} ]$ we have:

$$[\phi {\kern 1pt} (a_{\alpha }^{1} ),\phi {\kern 1pt} (a_{\alpha }^{2} )] \subset \text{R}$$

(6.42)

and for l ∈ R:

$$[\phi {\kern 1pt} (a_{\alpha }^{1} + l),\phi {\kern 1pt} (a_{\alpha }^{2} + l)] \subset \text{R}.$$

(6.43)

If l is the value of the random variable L, the lower and upper bounds of (6.43) depend only on l for a given level α. The mathematical expectation for each bound is now computed:

$$E[\phi {\kern 1pt} (a_{\alpha }^{1} + l),\phi {\kern 1pt} (a_{\alpha }^{2} + l)] = \left[ {\int\limits_{{l_{1} }}^{{l_{2} }} \phi (a_{\alpha }^{1} + l){\kern 1pt} \cdot {\kern 1pt} f(l)dl,\int\limits_{{l_{1} }}^{{l_{2} }} \phi (a_{\alpha }^{2} + l){\kern 1pt} \cdot {\kern 1pt} f(l)dl} \right].$$

(6.44)

Theorem

(Kaufmann and Gupta 1991). The membership function of the mathematical expectation of a hybrid number $(\tilde{A},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} )$ is the membership of $\tilde{A}$ shifted by the mathematical expectation of $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}$

Proof

Using the intervals of confidence of level α:

$$\begin{aligned} E_{\alpha } (\tilde{A}[ + ]\underline{L} ) & = \left[ {\int\limits_{{l_{1} }}^{{l_{2} }} (a_{\alpha }^{1} + l){\kern 1pt} \cdot f(l)dl,\int\limits_{{l_{1} }}^{{l_{2} }} (a_{\alpha }^{2} + l){\kern 1pt} \cdot f(l)dl} \right] \\ & = \left[ {a_{\alpha }^{1} {\kern 1pt} \cdot {\kern 1pt} \int\limits_{{l_{1} }}^{{l_{2} }} f(l)dl{\kern 1pt} + {\kern 1pt} \int\limits_{{l_{1} }}^{{l_{2} }} l{\kern 1pt} \cdot {\kern 1pt} f(l)dl,a_{\alpha }^{2} {\kern 1pt} \cdot {\kern 1pt} \int\limits_{{l_{1} }}^{{l_{2} }} f(l)dl{\kern 1pt} + {\kern 1pt} \int\limits_{{l_{1} }}^{{l_{2} }} l{\kern 1pt} \cdot {\kern 1pt} f(l)dl} \right] \\ & = \left[ {a_{\alpha }^{1} + E(l){\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} a_{\alpha }^{2} + E(l)} \right] \\ \end{aligned}$$

(6.45)

Hence, in a hybrid sum, if the random variables satisfy their random expectation, they will have the same effect as ordinary numbers, shifting the sum of fuzzy numbers.

Using the notation $({\tilde {A}},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} ) = {\tilde{A}}( + )^{\prime} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L} ,$ where $\tilde{A}$ is a triangular fuzzy number, the following example is illustrated.

Example

Let $\tilde{A}_{1} = (3,5,9)( + )^{\prime}(6,1.2)$ and $\tilde{A}_{2} = (6,7,10)( + )^{\prime}(7,1.8)$ be two hybrid numbers, then

$$\begin{aligned} \underline{{\tilde{A}}}_{1} \oplus \underline{{\tilde{A}}}_{2} & = [(3,5,9)( + )^{\prime}(6,1.2)] \oplus [(6,7,10)( + )^{\prime}(7,1.8)] \\ & = [(9,11,15)( + )^{\prime}(0,1.2)] \oplus [(13,14,17)( + )^{\prime}(0,1.8)] \\ & = (22,25,32)( + )^{\prime}(0,3.0). \\ \end{aligned}$$