A Fuzzy Inventory Model for a Deteriorating Item with Variable Demand, Permissible Delay in Payments and Partial Backlogging with Shortage Follows Inventory (SFI) Policy

Abstract

This research studies a fuzzy inventory model for a deteriorating item with permissible delay in payments. For this paper, the demand depends on selling price and the frequency of the advertisement. In order to make a more realistic inventory model, it is considered the case of stock-out which is partial backlogged. In this work, it is taken account the shortage follows inventory (SFI) policy. Several scenarios and sub-scenarios have been provided, and each corresponding problem has been defined as a constrained optimization problem in the fuzzy environment. Further, these problems have converted into a new problem using the nearest interval approximation technique of fuzzy numbers. Quantum-behaved particle swarm optimization (QPSO) algorithm with the help of interval mathematics has been used to solve the optimization problems. Numerical examples have been solved in order to illustrate the proposed inventory model. Finally, with the aim to analyse the significant influence of different factors on the optimal policies, a sensitivity analysis is done.

Appendices

Appendix A: Brief Description of Quantum-Behaved Particle Swarm Optimization (QPSO)

This Appendix is briefly discussing about the efficient soft computing method named as quantum-behaved particle swarm optimization method (see for instance Bhunia and Shaikh [59]). The solution found by this QPSO method is known as efficient solution or in other words can be termed as optimal answer. Unfortunately, the optimality of the solution cannot be proved theoretically. It is well known that Kennedy and Eberhart [60] developed the particle swarm optimization technique considering generic behaviour of bird flocking/fish schooling.

In particle swarm optimization, the different attributes of ith ($1 \le i \le p_{\text{size}}$) particles are as follows:

• 1. $x_{i} = (x_{i1} ,x_{i2} , \ldots ,x_{in} )$ is current position in search spaces.

• 2. $v_{i} = (v_{i1} ,v_{i2} , \ldots ,v_{in} )$ is current velocity.

• 3. $p_{i} = (p_{i1} ,p_{i2} , \ldots ,p_{in} )$ is personal best position or (pbest)

• 4. $p_{\text{g}} = (p_{{{\text{g}}1}} ,p_{{{\text{g}}2}} , \ldots ,p_{{{\text{g}}n}} ).$ is global best (gbest) position

According to Kennedy and Eberhart [60], the velocity of ith particle in kth iteration/generation is updated by the following rule:

$$v_{ij}^{(k + 1)} = wv_{ij}^{(k)} + c_{1} r_{1j}^{(k)} \left( {p_{ij}^{(k)} - x_{ij}^{(k)} } \right) + c_{2} r_{2j}^{(k)} \left( {p_{gj}^{(k)} - x_{ij}^{(k)} } \right),\quad {\text{where}}\;j = 1,2, \ldots ,n,\;k = 1,2, \ldots ,m_{g}$$

Here, inertia weight is $w,$ acceleration coefficients are $c_{1} \& c_{2}$ & $r_{1j}^{(k)} \sim U(0, \, 1); \, r_{2j}^{(k)} \sim U(0, \, 1).$

At (k + 1)-th iteration, the new position of ith particle is calculated by

$$x_{ij}^{(k + 1)} = x_{ij}^{(k)} + v_{ij}^{(k + 1)} \quad {\text{i}} . {\text{e}} . ,\quad x_{i}^{(k + 1)} = x_{i}^{(k)} + v_{i}^{(k + 1)}$$

The personal best (pbest) position of i-th particle is updated as follows:

$$p_{i}^{(k + 1)} = \left\{ {\begin{array}{*{20}l} {p_{i}^{(k)} } \hfill & {{\text{if}}\; \, f\left( {x_{i}^{(k + 1)} } \right) \le \,f\left( {p_{i}^{(k)} } \right)} \hfill \\ {x_{i}^{(k)} } \hfill & {{\text{if}}\;f\left( {x_{i}^{(k + 1)} } \right) > f\left( {p_{i}^{(k)} } \right)} \hfill \\ \end{array} } \right.\quad {\text{with}}\;p_{i}^{(0)} = x_{i}^{(0)}$$

where the objective is to maximize function f.

Global best (gbest) position p _g can be identified by any particle at the time of all previous iterations/generations is defined as $p_{g}^{(k + 1)} = \arg \max_{{p_{i} }} f(p_{i}^{(k + 1)} ),\quad 1 \le i \le p_{\text{size}} .$

Each particle must converge to its local attractor $\tilde{p}_{i} = (\tilde{p}_{i1} ,\tilde{p}_{i2} , \ldots ,\tilde{p}_{in} )$ (Clerc and Kennedy [61]) whose components are given by

$$\begin{aligned} & \tilde{p}_{ij}^{(k)} = {{\left[ {c_{1} p_{ij}^{(k)} + c_{2} p_{gj}^{(k)} } \right]} \mathord{\left/ {\vphantom {{\left[ {c_{1} p_{ij}^{(k)} + c_{2} p_{gj}^{(k)} } \right]} {\left( {c_{1} + c_{2} } \right)}}} \right. \kern-0pt} {\left( {c_{1} + c_{2} } \right)}},\quad j = 1,2, \ldots ,n \\ & {\text{or}}\;\tilde{p}_{ij}^{(k)} = \phi_{j} p_{ij}^{(k)} + \left( {1 - \phi_{j} } \right)p_{gj}^{(k)} ,\quad j = 1,2, \ldots ,n \\ \end{aligned}$$

where $\phi_{j} = \frac{{c_{1} r_{1j}^{(k)} }}{{c_{1} r_{1j}^{(k)} + c_{2} r_{2j}^{(k)} }}$where $\phi_{j} \sim U(0,1).$

After Kennedy and Eberhart [60], some new variants of PSO have been proposed considering different velocity updating rules. Amongst these, the popular versions of PSO are: (1) weighted PSO (Clerc [62]) and (2) PSO-CO (Clerc and Kennedy [61]). It is worth mentioning that in these versions of PSO, particle’s behaviour is according to the rule of classical mechanics; here its position and velocity vectors only depict a particle in the swarm. Nevertheless, this is not true in quantum mechanics. Considering quantum behaviour of particle, Sun et al. [63, 64] introduced quantum-behaved PSO (QPSO). According to Sun et al. [63, 64], the iterative equation for the position of the particle in QPSO is determined by

$$x_{ij}^{(k)} = \tilde{p}_{ij}^{(k)} \pm \beta^{{\prime }} \left| {m_{j}^{(k)} - x_{ij}^{(k)} } \right|\log \left( {\frac{1}{{u_{j}^{(k)} }}} \right)$$

where $u_{j}^{(k)} \sim U(0,1)$ and $\beta^{{\prime }}$ is acted as contraction–expansion coefficient which worked in order to control convergence speed of an algorithm. The value of $\beta^{{\prime }}$ can be decreased linearly from 1.0 to 0.5. The global point is called as mainstream (or mean best $(m^{(k)} )$) of the population at k-th iteration is the mean of the pbest positions of each and every particles.

$$m^{(k)} = \left( {m_{1}^{(k)} ,m_{2}^{(k)} , \ldots ,m_{n}^{(k)} } \right) = \left( {\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {p_{i1}^{(k)} ,\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {p_{i2}^{(k)} , \ldots ,\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {p_{in}^{(k)} } } } } \right).$$

In WQPSO, weighted mean best position is replacing mean best position of QPSO. Therefore, particles can be ranked in decreasing order as per their fitness values. Further, weighted coefficient $\alpha_{i}$ is assigned and linearly decreasing with the particle’s rank such that nearer the best solution, the larger its weighted coefficient is. The mean best position $m^{(k)}$, therefore, is calculated with:

$$m^{(k)} = \left( {m_{1}^{(k)} ,m_{2}^{(k)} , \ldots ,m_{n}^{(k)} } \right) = \left( {\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {\alpha_{i1} p_{i1}^{(k)} ,\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {\alpha_{i2} p_{i2}^{(k)} , \ldots ,\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {\alpha_{in} p_{in}^{(k)} } } } } \right)$$

where $\alpha_{i}$ is the weighted coefficient and $\alpha_{\text{id}}$ is the dimension coefficient of every particle. In this work, the weighted coefficient for each particle decreases linearly from 1.5 to 0.5.

On the hand, in GQPSO, $\tilde{p}_{ij}^{(k)}$ is calculated as follows:

$$\tilde{p}_{ij}^{(k)} = {{\left[ {G^{(k)} p_{ij}^{(k)} + g^{(k)} p_{gj}^{(k)} } \right]} \mathord{\left/ {\vphantom {{\left[ {G^{(k)} p_{ij}^{(k)} + g^{(k)} p_{gj}^{(k)} } \right]} {\left( {G^{(k)} + g^{(k)} } \right)}}} \right. \kern-0pt} {\left( {G^{(k)} + g^{(k)} } \right)}},\quad j = 1,2, \ldots ,n$$

where $G^{(k)}$ and $g^{(k)}$ be the random numbers (at kth iteration) which are generated using the absolute value of the Gaussian (Normal) probability distribution with mean (0) and variance (1).

Here $m^{(k)}$ is computed by

$$m^{(k)} = \left( {m_{1}^{(k)} ,m_{2}^{(k)} , \ldots ,m_{n}^{(k)} } \right) = \left( {\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {p_{i1}^{(k)} ,\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {p_{i2}^{(k)} , \ldots ,\frac{1}{{p_{\text{size}} }}\sum\limits_{i = 1}^{{p_{\text{size}} }} {p_{in}^{(k)} } } } } \right)$$

and the iterative equation for the position of the particle is given by

$$x_{ij}^{(k)} = \tilde{p}_{ij}^{(k)} \pm \beta^{{\prime }} \left| {m_{j}^{(k)} - x_{ij}^{(k)} } \right|\log \left( {\frac{1}{{G^{(k)} }}} \right)$$

where $\beta^{{\prime }}$ decreases linearly from 1.0 to 0.5.

Appendix B: Interval Arithmetic and Order Relations

A real number A is represented as an interval number with the form $A = \left[ {a_{\text{L}} ,a_{\text{R}} } \right] = \left\{ {x:a_{\text{L}} \le x \le a_{\text{R}} ,x \in {\mathbb{R}}} \right\}$ of the width (a_R − a_L). So, each real number $x \in {\mathbb{R}}$ is represented as an interval number [x, x] with zero width. In the other way, an interval number can be represented with the centre and radius form as follows $A = \left\langle {a_{\text{C}} ,a_{\text{W}} } \right\rangle = \left\{ {x:a_{\text{C}} {-}a_{\text{W}} \le x \le a_{\text{C}} + a_{\text{W}} ,x \in {\mathbb{R}}} \right\}$, where centre $a_{\text{C}} = (a_{\text{L}} + a_{\text{R}} )/2$ and ${\text{radius}} = a_{\text{W}} = \left( {a_{\text{R}} {-}a_{\text{L}} } \right)/2.$

Definition 1

Let $A = \, \left[ {a_{\text{L}} ,a_{\text{R}} } \right]$ and $B = \left[ {b_{\text{L}} ,b_{\text{R}} } \right]$ be two interval numbers. Then addition of two interval numbers, subtraction of two interval numbers, multiplication with scalar number, multiplication of two interval numbers and division of two interval numbers are described below:

Addition of two interval numbers: $A + B = \left[ {a_{\text{L}} ,a_{\text{R}} } \right] + \left[ {b_{\text{L}} ,b_{\text{R}} } \right] = \left[ {a_{\text{L}} + b_{\text{L}} ,a_{\text{R}} + b_{\text{R}} } \right].$

Subtraction two interval numbers: $A - B = \left[ {a_{\text{L}} , a_{\text{R}} } \right] = \left[ {b_{\text{L}} ,b_{\text{R}} } \right] = \left[ {a_{\text{L}} ,a_{\text{R}} } \right] + \left[ { - \,b_{\text{R}} , - \,b_{\text{L}} } \right] = \left[ {a_{\text{L}} - b_{\text{R}} ,a_{\text{R}} - b_{\text{L}} } \right].$

Multiplication with scalar number:

$$\lambda A = \lambda \left[ {a_{\text{L}} , \, a_{\text{R}} } \right] = \left\{ {\begin{array}{*{20}l} {\left[ {\lambda a_{\text{L}} , \, \lambda a_{\text{R}} } \right]} \hfill & {{\text{if}}\;\lambda \ge 0} \hfill \\ {\left[ {\lambda a_{\text{R}} , \, \lambda a_{\text{L}} } \right]} \hfill & {{\text{if}}\;\lambda < 0} \hfill \\ \end{array} } \right.$$

Multiplication of two interval numbers: $A \times B = \left[ {\hbox{min} \left( {a_{\text{L}} b_{\text{L}} ,a_{\text{L}} b_{\text{R}} ,a_{\text{R}} b_{\text{L}} ,a_{\text{R}} b_{\text{R}} } \right),\hbox{max} \left( {a_{\text{L}} b_{\text{L}} ,a_{\text{L}} b_{\text{R}} ,a_{\text{R}} b_{\text{L}} ,a_{\text{R}} b_{\text{R}} } \right)} \right]$

Division of two interval numbers:

$$\frac{A}{B} = A \times \left( {\frac{1}{B}} \right) = \left[ {a_{\text{L}} , \, a_{\text{R}} } \right] \times \left[ {\frac{1}{{b_{\text{R}} }}, \, \frac{1}{{b_{\text{L}} }}} \right]\quad {\text{where}}\quad 0 \notin \left[ {b_{\text{L}} ,b_{\text{R}} } \right]$$

Definition 2

Let us consider $A = \left\langle {a_{\text{C}} ,a_{\text{W}} } \right\rangle$ and $B = \left\langle {b_{\text{C}} ,b_{\text{W}} } \right\rangle$ in the form of centre and radius. Then, addition, subtraction and scalar multiplication of interval numbers with centre and radius form are described as follows:

$$\begin{aligned} & A + B = \left\langle {a_{\text{C}} + b_{\text{C}} ,a_{\text{W}} + b_{\text{W}} } \right\rangle \\ & A{-}B = \, \left\langle {a_{\text{C}} ,a_{\text{W}} } \right\rangle - \left\langle {b_{\text{C}} ,b_{\text{W}} } \right\rangle = \left\langle {a_{\text{C}} ,a_{\text{W}} } \right\rangle + \left\langle { - b_{\text{C}} ,b_{\text{W}} } \right\rangle = \left\langle {a_{\text{C}} {-}b_{\text{C}} ,a_{\text{W}} + b_{\text{W}} } \right\rangle \\ & \lambda A = \lambda \left\langle {a_{\text{C}} ,a_{\text{W}} } \right\rangle = \left\langle {\lambda a_{\text{C}} ,\lambda a_{\text{W}} } \right\rangle \\ \end{aligned}$$

2.1 Interval Order Relations

Consider $A = [a_{\text{L}} ,a_{\text{R}} ]$ and $B = [b_{\text{L}} ,b_{\text{R}} ]$ are two interval numbers. There may be any one of the form happened of these two intervals which describe as follows:

Case 1:

These two intervals are distinct and disjoint (see Fig. 2).

Fig. 2

Type-1 intervals

Case 2:

These two intervals are partially related or overlapping (see Fig. 3).

Fig. 3

Type-2 intervals

Case 3:

Any one of the intervals contains other (see Fig. 4).

Fig. 4

Type-3 intervals

Sahoo et al. [65] have proposed the definitions of interval order relations between two interval numbers in order to solve the maximization and minimization problems.

Definition 3

Order relation of type $>_{\hbox{max} }$ between two intervals. Let us consider two intervals $A = [a_{\text{L}} ,a_{\text{R}} ] = \left\langle {a_{\text{c}} ,a_{\text{w}} } \right\rangle$ and $B = [b_{\text{L}} ,b_{\text{R}} ] = \left\langle {b_{\text{c}} ,b_{\text{w}} } \right\rangle$. Therefore, for solving the maximization problems, the following properties hold:

• 1. $A >_{\hbox{max} } B \Leftrightarrow a_{\text{c}} > b_{\text{c}} \;{\text{for}}\;{\text{Type}}\;{\text{I}}\;{\text{and}}\;{\text{Type}}\;{\text{II}}\;{\text{intervals}},$

• 2. $A >_{\hbox{max} } B \Leftrightarrow$ either $a_{\text{c}} \ge b_{\text{c}} \wedge a_{\text{w}} < b_{\text{w}}$ or $a_{\text{c}} \ge b_{\text{c}} \wedge a_{\text{R}} > b_{\text{R}} \;{\text{for}}\;{\text{Type}}\;{\text{III}}\;{\text{intervals}},$

According to the above definition, the interval number $A$ is accepted for maximization case. So, the order relation $A >_{\hbox{max}} B$ is not symmetric, but it is reflexive and transitive.

Definition 4

Order relation of type $<_{\hbox{min} }$ between two intervals. Let us consider two interval numbers $A = [a_{\text{L}} ,a_{\text{R}} ] = \left\langle {a_{\text{c}} ,a_{\text{w}} } \right\rangle$ and $B = [b_{\text{L}} ,b_{\text{R}} ] = \left\langle {b_{\text{c}} ,b_{\text{w}} } \right\rangle$. Then, for solving the minimization problems, the following properties hold:

• 1. $A <_{\hbox{min} } \; B \Leftrightarrow a_{\text{c}} < b_{\text{c}} \;{\text{for}}\;{\text{Type}}\;{\text{I}}\;{\text{and}}\;{\text{Type}}\;{\text{II}}\;{\text{intervals}},$

• 2. $A <_{\hbox{min} } \; B \Leftrightarrow$ either $a_{\text{c}} \le b_{\text{c}} \wedge a_{\text{w}} < b_{\text{w}}$ or $a_{\text{c}} \le b_{\text{c}} \wedge a_{\text{L}} < b_{\text{L}} \;{\text{for}}\;{\text{Type}}\;{\text{III}}\;{\text{intervals}},$

According to the above definition, the interval $A$ is accepted for minimization case. So, the order relation $A \,{<_{\hbox{min} }}\, B$ is reflexive and transitive, but it is not symmetric.

$A \,{<_{\hbox{min} }}\, B$ is reflexive and transitive, but it is not symmetric.