A two-warehouse inventory model for non-instantaneous deteriorating items with interval-valued inventory costs and stock-dependent demand under inflationary conditions

Abstract

This research work develops a two-warehouse inventory model for non-instantaneous deteriorating items with interval-valued inventory costs and stock-dependent demand under inflationary conditions. The proposed inventory model permits shortages, and the backlogging rate is variable and dependent on the waiting time for the next order, and inventory parameters are interval-valued. The main aim of this research is to obtain the retailer’s optimal replenishment policy that minimizes the present worth of total cost per unit time. The optimization problems of the inventory model have been formulated and solved using two variants of particle swarm optimization (PSO) and interval order relations. The efficiency and effectiveness of the inventory model are validated with numerical examples and a sensitivity analysis. The proposed inventory model can assist a decision maker in making important replenishment decisions.

Appendices

Appendix 1

This appendix presents a brief overview of interval arithmetic.

The brief summary of interval mathematics is from Hansen and Walster [53]. The definitions of interval arithmetic, interval functions, interval order relations and central tendencies of interval numbers that are indispensable in the formulation of the two-warehouse inventory model are given below.

An interval number A is defined as $A = \left[ {a_{L} ,a_{R} } \right] = \, \left\{ {x:a_{L} \le x \le a_{R} ,x \in {\mathbb{R}}} \right\}$ of width $\left( {a_{R} - a_{L} } \right)$. Here, ${\mathbb{R}}$ is the set of real numbers.

Every real number $x \in {\mathbb{R}}$ is expressed as a degenerate interval number [x, x] with zero width. Interval number is also written in the form of centre and radius of the interval as A$= \left\langle {a_{C} , \, a_{W} } \right\rangle = \left\{ {x:a_{C} {-}a_{W} \le x \le a_{C} + a_{W} , \, x \in {\mathbb{R}}} \right\},\quad {\text{where}}\;a_{C} = \, \left( {a_{L} + a_{R} } \right)/2 = {\text{center}}\;{\text{of}}\;{\text{the}}\;{\text{interval}}\;{\text{and}}\;a_{W} = \left( {a_{R} {-}a_{L} } \right)/2 = {\text{radius}}\;{\text{of}}\;{\text{the}}\;{\text{interval}}.$

Definition 1

Let $A = [a_{L} ,a_{R} ]$ and $B = [b_{L} ,b_{R} ]$ be two interval numbers. The addition, subtraction, scalar multiplication, multiplication and division of interval numbers are given below:

• 1. Addition: $A + B \, = \left[ {a_{L} ,a_{R} } \right] + \left[ {b_{L} ,b_{R} } \right] = \left[ {a_{L} + \, b_{L} ,a_{R} + \, b_{R} } \right].$

• 2. Subtraction: $A \, - \, B = \, \left[ {a_{L} , a_{R} } \right] - \left[ {b_{L} ,b_{R} } \right] = \left[ {a_{L} , a_{R} } \right] + \left[ { - b_{L} , - b_{R} } \right] = \left[ {a_{L} - b_{L} ,a_{R} - b_{R} } \right].$

• 3. Scalar multiplication:$\lambda A = \lambda \left[ {a_{L} , \, a_{R} } \right] = \left\{ {\begin{array}{*{20}l} {\left[ {\lambda a_{L} , \, \lambda a_{R} } \right]} & {\quad {\text{if}}\;\lambda \ge 0} \\ {\left[ {\lambda a_{R} , \, \lambda a_{L} } \right]} & {\quad {\text{if}}\;\lambda < 0,} \\ \end{array} } \right.\quad {\text{for}}\;{\text{any}}\;{\text{real}}\;{\text{number}}\;\lambda .$

• 4. Multiplication: $A \times B = \, \left[ {\min \, \left( {a_{L} b_{L} ,a_{L} b_{R} ,a_{R} b_{L} ,a_{R} b_{R} } \right), \, \max\left( {a_{L} b_{L} ,a_{L} b_{R} ,a_{R} b_{L} ,a_{R} b_{R} } \right)} \right].$

• 5. Division: $\frac{A}{B} = A \times \left( {\frac{1}{B}} \right) = \left[ {a_{L} , \, a_{R} } \right] \times \left[ {\frac{1}{{b_{R} }}, \, \frac{1}{{b_{L} }}} \right]$ provided 0 ∉ [b_L, b_R].

Interval order relations

Let $A = [a_{L} ,a_{R} ]$ and $B = [b_{L} ,b_{R} ]$ be two intervals. Then, these two intervals can be anyone of the following types:

• Type 1: Two intervals are disjoint.

• Type 2: Two intervals are partially overlapping.

• Type 3: One of the intervals contains the other one.

Some researchers gave the definitions of order relations between two interval numbers. In recent times, Sahoo et al. [54] provided the same by modifying the drawbacks of existing definitions. Their definitions are as follows:

Definition 2

The order relation $>_{\hbox{max} }$ among the intervals $A = [a_{L} ,a_{R} ] = \left\langle {a_{c} ,a_{w} } \right\rangle$ and $B = [b_{L} ,b_{R} ] = \left\langle {b_{c} ,b_{w} } \right\rangle$; then, for maximization problems

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

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

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

Definition 3

The order relation $<_{\hbox{min} }$ among the intervals $A = [a_{L} ,a_{R} ] = \left\langle {a_{c} ,a_{w} } \right\rangle$ and $B = [b_{L} ,b_{R} ] = \left\langle {b_{c} ,b_{w} } \right\rangle$; then, for minimization problems

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

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

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

Appendix 2

This appendix completes the modelling part of case when $t_{{\rm d}} \ge t_{{\rm r}}$. In this case, time during which no deterioration happens is greater than the time during which inventory in RW attains zero and the behaviour of the inventory model over the entire cycle $\left[ {0, \, T} \right]$ is graphically depicted in Fig. 2.

Fig. 2

Two-warehouse inventory system when $t_{{\rm d}} > t_{{\rm r}}$

The differential equations that model the inventory level in the RW and OW at any time t within the period $\left( {0, \, T} \right)$ are expressed as:

$$\frac{{{\text{d}}I_{{\rm r}} ( t )}}{{{\text{d}}t}} = - \left( {a \, + \, bI_{{\rm r}} ( t )} \right),\quad 0 \le t \le t_{{\rm r}}$$

(22)

$$\frac{{{\text{d}}I_{0} ( t )}}{{{\text{d}}t}} = - \left( {a \, + \, bI_{0} ( t )} \right),\quad t_{{\rm r}} < t \le t_{{\rm d}}$$

(23)

$$\frac{{{\text{d}}I_{0} ( t )}}{{{\text{d}}t}} + \alpha I_{0} ( t ) = - \left( {a \, + \, bI_{0} ( t )} \right),\quad t_{{\rm d}} < t \le t_{{\rm w}}$$

(24)

$$\frac{{{\text{d}}B( t )}}{{{\text{d}}t}} = ae^{ - \delta (T - t)} ,\quad t_{{\rm w}} < t \le T$$

(25)

The solutions of the above differential equations with boundary conditions $I_{{\rm r}} \left( {t_{{\rm r}} } \right) = 0,I_{0} \left( {t_{{\rm r}} } \right) = W,I_{0} \left( {t_{W} } \right) = 0,B\left( {t_{W} } \right) = 0$ are

$$I_{{\rm r}} ( t ) = \frac{a}{b}\left( {e^{{b\left( {t_{{\rm r}} - t} \right)}} - 1} \right),\quad 0 \le t \le t_{{\rm r}}$$

(26)

$$I_{0} ( t ) = \left( {W + \frac{a}{b}} \right)e^{{b\left( {t_{{\rm r}} - t} \right)}} - \frac{a}{b},\quad t_{{\rm r}} < t \le t_{{\rm d}}$$

(27)

$$I_{0} ( t ) = \frac{a}{\alpha + b}\left( {e^{{\left( {\alpha + b} \right)\left( {t_{{\rm w}} - t} \right)}} - 1} \right),\quad t_{{\rm d}} < t \le t_{{\rm w}}$$

(28)

$$B( t ) = \frac{a}{\delta }\left\{ {e^{ - \delta (T - t)} - e^{{ - \delta (T - t_{{\rm w}} )}} } \right\},\quad t_{{\rm w}} < t \le T$$

(29)

The quantity of lost sales at time t is

$$L(t) = \int\limits_{{t_{{\rm w}} }}^{t} {a\left\{ {1 - e^{ - \delta (T - t)} } \right\}{\text{d}}t} ;\quad t_{{\rm w}} < t \le T$$

$$\, L(t) = a\left[ {\left( {t - t_{{\rm w}} } \right) - \frac{1}{\delta }\left\{ {e^{ - \delta (T - t)} - e^{{ - \delta (T - t_{{\rm w}} )}} } \right\}} \right]$$

(30)

In view that there exists continuity of $I_{0} (t)$ at $t = t_{{\rm d}} ,$, thus

$$\left( {W + \frac{a}{b}} \right)e^{{b\left( {t_{{\rm r}} - t_{{\rm d}} } \right)}} - \frac{a}{b} = \frac{a}{\alpha + b}\left( {e^{{\left( {\alpha + b} \right)\left( {t_{{\rm w}} - t_{{\rm d}} } \right)}} - 1} \right)$$

$$t_{{\rm w}} = t_{{\rm d}} + \frac{1}{\alpha + b}\ln \left| {1 + \frac{\alpha + b}{a}\left\{ {\left( {W + \frac{a}{b}} \right)e^{{b\left( {t_{{\rm r}} - t_{{\rm d}} } \right)}} - \frac{a}{b}} \right\}} \right|$$

(31)

Now, using $I_{{\rm r}} \left( 0 \right) = \, Z - W$, then the maximum inventory is

$$Z = W + \frac{a}{b}\left( {e^{{bt_{{\rm r}} }} - 1} \right)$$

(32)

By placing $t \, = \, T$ into Eq. (29), the maximum amount of demand backlogged per cycle is

$$B\left( T \right) = \frac{a}{\delta }\left( {1 - e^{{ - \delta (T - t_{{\rm w}} )}} } \right)$$

(33)

As a result, order quantity is $Q = Z + B(T)$

$$Q = W + \frac{a}{b}\left( {e^{{bt_{{\rm r}} }} - 1} \right) + \frac{a}{\delta }\left( {1 - e^{{ - \delta (T - t_{{\rm w}} )}} } \right)$$

(34)

Again, the total cost per cycle consists of the following elements:

• 1. Present worth of the replenishment cost = A

• 2. Present worth of the inventory holding cost in RW = $F\int\limits_{0}^{{t_{{\rm r}} }} {e^{ - rt} I_{{\rm r}} ( t ){\text{d}}t}$

  $$= \frac{Fa}{b}\left\{ {\frac{1}{{\left( {r + b} \right)}}\left( {e^{{bt_{{\rm r}} }} - e^{{ - rt_{{\rm r}} }} } \right) + \frac{1}{r}\left( {e^{{ - rt_{{\rm r}} }} - 1} \right)} \right\}$$

• 3. Present worth of the inventory holding cost in OW

  $$\begin{aligned} & = H\left( {\int\limits_{0}^{{t_{{\rm r}} }} {e^{ - rt} W{\text{d}}t} + \int\limits_{{t_{{\rm r}} }}^{{t_{{\rm d}} }} {e^{ - rt} I_{0} ( t ){\text{d}}t + \int\limits_{{t_{{\rm d}} }}^{{t_{{\rm w}} }} {e^{ - rt} I_{0} ( t ){\text{d}}t} } } \right) \\ & = H\left[ {\frac{W}{r}\left( {1 - e^{{ - rt_{{\rm r}} }} } \right) + \frac{{e^{{ - rt_{{\rm r}} }} }}{r + b}\left( {W + \frac{a}{b}} \right)\left( {1 - e^{{\left( {r + b} \right)\left( {t_{{\rm r}} - t_{{\rm d}} } \right)}} } \right) + \frac{{ae^{{ - rt_{{\rm r}} }} }}{br}\left( {e^{{r\left( {t_{{\rm r}} - t_{{\rm d}} } \right)}} - 1} \right)} \right. \\ & \quad \left. {\left. { + \frac{{ae^{{ - rt_{{\rm w}} }} }}{\alpha + b}\left\{ {\frac{1}{\alpha + b + r}\left( {e^{{\left( {\alpha + b + r} \right)\left( {t_{{\rm w}} - t_{{\rm d}} } \right)}} - 1} \right)} \right. + \frac{1}{r}\left( {1 - e^{{r\left( {t_{{\rm w}} - t_{{\rm d}} } \right)}} } \right)} \right\}} \right] \\ \end{aligned}$$

• 4. Present worth of the backlogging cost = $s\int\limits_{{t_{W} }}^{T} {B( t )e^{ - rt} {\text{d}}t}$

  $$= \frac{sa}{\delta }e^{ - \delta T} \left[ {\frac{1}{\delta - r}\left\{ {e^{(\delta - r)T} - e^{{(\delta - r)t_{{\rm w}} }} } \right\} + \frac{{e^{{\delta t_{{\rm w}} }} }}{r}\left\{ {e^{ - rT} - e^{{ - rt_{{\rm w}} }} } \right\}} \right]$$

• 5. Present worth of opportunity cost due to lost sales $= c_{1} e^{ - rT} \int\limits_{{t_{{\rm w}} }}^{T} {\left\{ {1 - e^{ - \delta (T - t)} } \right\}D{\text{d}}t}$

  $$= c_{1} ae^{ - rT} \left[ {T - t_{{\rm w}} - \frac{1}{\delta }\left\{ {1 - e^{{ - \delta (T - t_{{\rm w}} )}} } \right\}} \right]$$

• 6. Present worth of the cost for deteriorated items $= c\alpha \int\limits_{{t_{{\rm d}} }}^{{t_{{\rm w}} }} {e^{ - rt} I_{0} (t){\text{d}}t}$

  $$= c\frac{{ae^{{ - rt_{{\rm w}} }} }}{\alpha + b}\left\{ {\frac{1}{\alpha + b + r}\left( {e^{{\left( {\alpha + b + r} \right)\left( {t_{{\rm w}} - t_{{\rm d}} } \right)}} - 1} \right) + \frac{1}{r}\left( {1 - e^{{r\left( {t_{{\rm w}} - t_{{\rm d}} } \right)}} } \right)} \right\}$$

Appendix 3: Particle swarm optimization (PSO)

This appendix provides a brief overview of PSO. The concept of particle swarm optimization algorithm (PSO) was introduced by Eberhart and Kennedy [55] and Kennedy and Eberhart [56]. This algorithm has been used broadly in obtaining solutions for optimization problems. The basic theory of PSO is based on the food-searching activities of birds. After Eberhart and Kennedy [55] and Kennedy and Eberhart [56], a lot of work has been published in this field by many researchers and they developed different variants of PSO. In this paper, two types of PSO algorithms named as PSO-CO and WQPSO are applied. Clerc [57] and Clerc and Kennedy [58] proposed an improved version of PSO, and this version of PSO is known as PSO-CO, i.e. constriction coefficient-based PSO, while Xi et al. [59] introduced the weighted quantum particle swarm optimization (WQPSO).

Below, it is presented the notation used in the PSO algorithm.

Notation	Description
N	Dimensionality of the search space
p_size	Population size
m_gen	Maximum number of generations
$\chi$	Constriction factor
$c_{1} \left( { > 0} \right)$	Cognitive learning rate
$c_{2} \left( { > 0} \right)$	Social learning rate
$r_{1}$, $r_{2}$	Uniformly distributed random numbers lying in the interval [0, 1].
$v_{i}^{(k)}$	Velocity of ith particle at kth generation/iteration
$x_{i}^{(k)}$	Position of ith particle of population at kth generation
$p_{i}^{(k)}$	Best previous position of ith particle at kth generation
$p_{g}^{(k)}$	Position of the best particle among all the particles in the population

The algorithms of the PSO-CO and WQPSO are given below.

Algorithm of particle swarm optimization with constriction (PSO-CO)

Step 1. Set all PSO parameters and bounds of the decision variables.

Step 2. Set a population size of particles with random positions and velocities.

Step 3. Determine the fitness value of all particles.

Step 4. Save track of the locations where each individual has its highest fitness so far.

Step 5. Save track of the position with the global best fitness.

Step 6. Update the velocity of each particle by using the following equation: $v_{i}^{(k + 1)} = \chi \left[ {v_{i}^{(k)} + C_{1} r_{1} \left( {p_{i}^{(k)} - x_{i}^{(k)} } \right) + c_{2} r_{2} \left( {p_{g}^{(k)} - x_{i}^{(k)} } \right)} \right]$ where $\chi = \frac{2}{{\left| {2 - (c_{1} + c_{2} ) - \sqrt {(c_{1} + c_{2} )^{2} - 4(c_{1} + c_{2} )} } \right|}}$

Step 7. Update the position of each particle by using the following equation: $x_{i}^{(k + 1)} = x_{i}^{(k)} + v_{i}^{(k + 1)}$

Step 8. If the stop criterion is reached, go to Step 9, else go to Step 3.

Step 9. Report the position and fitness of global best particle.

Step 10. Stop.

Algorithm of weighted quantum particle swarm optimization (WQPSO)

Step 1. Set all PSO parameters and bounds for the decision variables.

Step 2. Set a population size of particles with random positions.

Step 3. Determine the fitness value of each particle.

Step 4. Update the mean best position using the following equation: $m^{(k)} = \left( {m_{1}^{(k)} ,m_{2}^{(k)} , \ldots ,m_{n}^{(k)} } \right) = \left( {\frac{1}{{p_{size} }}\sum\limits_{i = 1}^{{p_{size} }} {\alpha_{i1} \tilde{p}_{i1}^{(k)} } ,\frac{1}{{p_{size} }}\sum\limits_{i = 1}^{{p_{size} }} {\alpha_{i2} \tilde{p}_{i2}^{(k)} } , \ldots ,\frac{1}{{p_{size} }}\sum\limits_{i = 1}^{{p_{size} }} {\alpha_{in} \tilde{p}_{in}^{(k)} } } \right)$ where $\alpha_{i}$ is the weighted coefficient and $\alpha_{id}$ is the dimension coefficient of every particle.

Step 5. Compare each particle’s fitness with the particle’s pbest. Save better one as pbest.

Step 6. Compare current gbest position with earlier gbest position.

Step 7. Update the position of each particle using the following equation: $x_{ij}^{(k)} = \tilde{p}_{ij}^{(k)} \pm \beta^{\prime } \left| {m_{j}^{(k)} - x_{ij}^{(k)} } \right|\log \left( {\frac{1}{{u_{j} }}} \right)$

Step 8. If the stop criterion is met, go to Step 9, else go to Step 3.

Step 9. Report the position and fitness of global best particle.

Step 10. Stop.