Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment

Abstract

In this study, we explored a multi-item inventory model for non-instantaneous deteriorating items under inflation in fuzzy rough environment with multiple warehouse facilities, where one is an owned warehouse and others are rented warehouses with limited storage capacity. Due to a number of uncertainties in the environment, the various expenditures and coefficients are considered as a fuzzy rough type. The objective and constraints in fuzzy rough are made deterministic using Tr–Pos chance constrained technique. The demand of items is considered as stock dependent, and deterioration of items is assumed to be constant over time. The model allows shortages in owned warehouse subject to partial backlogging. The purpose of this study is to find the retailer’s optimal replenishment policies to maximize the total profit. To illustrate the proposed model and also test the validity of the same, a numerical example is solved using the Mathematica-8.0 software. Sensitivity analysis is also performed to study the impact of important parameters on system decision variables, and its implications are discussed.

1 Introduction

The traditional inventory models are mainly developed with the single storage facility. But, in some situation that needs to store a large stock the existing single warehouse [own warehouse (OW)] inventory models are unsuitable. Then for storing the excess units, one (or more) additional warehouse(s) (RW) is hired on rental basis. Since the holding cost in the rental warehouses is generally assumed to be higher than that in the OW, it will be economical to supply the goods stored in RW first to customers and then give items in OW to customers. Hartely (1976) was the first to proposed a two-warehouse (2WH) inventory model. Sarma (1983) extended Hartleys model and considered the fixed transportation cost independent of the quantity being transported from RW to OW. Goswami and Chaudhuri (1992) further developed the model by considering a linear demand and transportation cost depending on the quantity being transported. Pakkala and Achary (1992) further investigated the two-warehouse model for deteriorating items. Rong et al. (2008) introduced two-warehouse inventory models for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time. Lee and Hsu (2009) presented a two-warehouse production model for deteriorating inventory items with time-dependent demands. Jaggi et al. (2011) discussed a two-warehouse partial backlogging inventory model for deteriorating items with linear trend in demand under inflationary conditions. Two-warehouse inventory models were mainly based on an assumption that the RW has unlimited capacity. But it is not realistic, thus for storing the excess units, more additional warehouses is hired on rental basis. Since the holding cost in the rental warehouses is generally assumed to be higher than that in the owned warehouse(OW), it will be economical to supply the goods stored in RW first to customers and then give items in OW to customers. Zhou (2003) introduced a multi-warehouse inventory model for items with time-varying demand and shortages. Das et al. (2015) described a multi-warehouse partial backlogging inventory model for deteriorating items under inflation when a delay in payment is permissible. Several researchers such as (Zhou and Yang 2005; Das et al. 2012; Liang and Zhou 2011; Chung et al. 2009; Kumar and Chanda 2018; Chakrabarty et al. 2018; Jaggi et al. 2017; Chakraborty et al. 2018) and many others have worked in the area of two-warehousing under different scenarios.

Uncertainty such as randomness, fuzziness, roughness, etc., are common in any real-life problems. Many researchers have focused their work on those imprecise environments (Chakraborty et al. 2014, 2015a, 2016; Castillo 2018; Valdez et al. 2019; Olivas et al. 2017). However in a decision-making process, some situation occurs where both fuzziness and roughness exist at the same time. In such situation, fuzzy rough (Fu-Ro) variables are used to define the problem. Dubois and Prade (1990) first introduced the concept of fuzzy rough sets. After that, some scholars generalized the concept of fuzzy rough sets (Radzikowska and Kerre 2002; Morsi and Yakout 1998). Xu and Zhao (2008) introduced a class of fuzzy rough expected value multi-objective decision-making model and its application to inventory problems. Xu and Zhao (2010) developed a multi-objective decision making with fuzzy rough coefficients and its application to the inventory to the inventory problem. A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money was studied by Mondal et al. (2013). Jana et al. (2014) developed a multi-objective multi-item inventory control problem in fuzzy rough environment. Manna et al. (2014) worked on three-layer supply chain in an imperfect production inventory model with two storage facilities under fuzzy rough environment. A fuzzy rough economic order quantity model for deteriorating items considering quantity discount and prepayment was developed by Taleizadeh et al. (2013).

In recent years, the research in inventory problems for deteriorating has been widely studied under various circumstances (Wee 1995; Widyadana et al. 2011; Lee and Dye 2012; Guchhait et al. 2014; Chakraborty et al. 2015b; Bhunia and Maiti 1998; Chakraborty et al. 2017). Ghare and Schrader (1963) were the first to consider the effect of decaying inventory under exponential decay. Covert and Philip (1973) extended this model by assuming deteriorating rate with two-parameter Weibull distribution. Philip (1974) further generalized the inventory model with a three-parameter Weibull distribution deterioration rate. In reality, there are many goods that do not deteriorate immediately and maintain fresh quality during a span of time. This phenomenon is termed as non-instantaneous deterioration. Wu et al. (2006) first presented this phenomenon and established an optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. Tiwari et al. (2017) developed two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization. Further, several researchers such as Shaikh et al. (2017), Ouyang et al. (2006), Chang et al. (2010), Valliathal and Uthayakumar (2011), Soni (2013), Maihami and Karimi (2014), Jaggi et al. (2015), Jaggi and Verma (2010), and Soni and Suthar (2018) have investigated inventory models for non-instantaneous deteriorating items under different environment.

In addition, due to high inflation rate, many developing countries like India, Bangladesh, Brazil, etc., the financial condition is being continuously changed and so it is not possible to ignore the effect of inflation. Nowadays, inflation has become a permanent feature of the economy throughout the world. Buzacott (1975) first introduced economic order quantity model with constant demand under the impact of inflation. Misra (1979) studied the effects of inflationary condition on inventory systems. Brahmbhatt (1982) also developed an EOQ model under variable rate of inflation. Hwang and Sohn (1983) established model for management of deteriorating inventory under inflation. An inventory model for deteriorating items with time-proportional demand and shortages under inflation have been discussed by Chen (1998). Wu (2003) investigated the effect of inflation and time discounting on inventory replenishment model for deteriorating item with time-varying demand and shortages in all cycles. Das et al. (2010) improved production policy for a deteriorating item under permissible delay in payments with stock-dependent demand rate. Guria et al. (2013) studied inventory policy for an item with inflation-induced purchasing price, selling price and demand with immediate part payment. Bhunia and Shaikh (2016) investigated two-warehouse (2WH) inventory problems in an interval environment under inflation. Recently, Chakrabarty et al. (2017) developed a production inventory model for defective items with shortages incorporating inflation and time value of money. The major assumption used in the related research articles is summarized in Table 1.

Table 1 Summary of related literature for warehouse inventory models

The rest of this paper is organized as follows: Sect. 2 defines the notations and assumption used throughout in this paper. In Sect. 3, we recall some preliminary knowledge about fuzzy rough variables and its application. Section 4 provides the mathematical formulation of multi-warehouse multi-item inventory system (MWMIIS). In Sect. 5, MWMIIS under fuzzy rough environment has been considered. Section 6 discusses the equivalent deterministic representation of the proposed multi-warehouse (nWH) multi-item fuzzy rough (MWMIFR) inventory model. Section 7 illustrates the proposed MWMIFR inventory model with a numerical example. Sections 7 and 8 furnish the numerical and graphical representations of the effect of different parameters. The managerial insights are discussed in Sect. 9. Section 10 summarizes the paper and also discusses about the scope of future work.

2 Notations and assumptions

2.1 Notations

For convenience, the following notations are used throughout the entire paper.

• $m=$ number of items.

• $k=$ number of warehouses.

• $W_{ji}=$ the fixed storage capacity of jth ($j=1, 2, 3, \ldots , k$) warehouse for ith item.

• $H_{j}=$ unit holding cost per unit item per unit time for jth ($j=1, 2, 3, \ldots , k$) warehouse.

• $D_i(t)=$ demand rate at time t for ith item.

• $\delta _i(t)=$ the backlogging rate for ith item which is a decreasing function of the waiting time t, without loss of generality, we here assume that $\delta _i(t)=e^{-\sigma _i t}$ where $\sigma _i\ge 0$, and t is waiting time.

• $\theta _i=$ deterioration rate of the ith item.

• $r=$ the discount rate.

• $f=$ the inflation rate, which is varied by the social economical situations.

• $R=(r-f)=$ representing the net discount rate of inflation.

• $B_{i}(t)=$ backlogged level at any time t(units) for ith item.

• $c^{l}_{i}=$ unit opportunity cost due to lost sale for ith item, if the shortage is lost.

• $\text {SC}_i=$ fixed cost of placing an order for ith item.

• $t^{d}_i=$ time period during which no deterioration occurs for ith item (time units).

• $c^{b}_{i}=$ unit backlogging cost per unit item per unit time for ith item, if the shortage is backlogged.

• $s_{i}=$ unit selling price for ith item.

• $p_i=$ unit purchasing cost for ith item.

• $\text {TC}_i=$ total transportation cost for ith item.

• $\text {TT}^{k-j+1}=$ total units of ith item transferred from $(k-j+ 1)$th ($j = 1, 2, 3, \ldots , k-1$) to 1st warehouse.

• $\text {TCP}^{k-j+1}_{i}=$ the transportation cost per unit per unit distance from $(k-j+1)$th $(j=1,2,3,\ldots ,k-1)$ to 1st warehouse for ith item.

• $d_{k-j+1}=$ distance of $(k-j+1)$th warehouse from 1st warehouse.

• $\text {TC}^{1}_{i}=$ fixed transportation cost for ith item.

• $\text {TP}_i=$ the profit per unit time for ith item

• $A_i=$ required storage area per unit quantity for ith item.

• $A=$ total available storage space.

• $B=$ available total budget cost.

Decision variables

• $T_i=$ the length of the replenishment cycle of ith item.

• $T_{ji}=$ the time at which the inventory level reaches zero for $(k-j+1)$th ($j = 1, 2, 3, \ldots , k$) (time unit) warehouse for ith item.

2.2 Assumptions

The mathematical model of the multi-warehouse multi-item inventory problem is based on the following assumptions:

• (i) Demand rate $D_i(t)$ at time t for ith item is

  $$\begin{aligned} D_{i}(t)=\left\{ \begin{array}{ll} \alpha _i+\beta _iI_{ji}(t), &{}\quad {I_{i}(t)> 0;} \\ \alpha _i, &{} \quad {I_{i}(t)\le 0.} \end{array} \right. \end{aligned}$$

  where $\alpha _i$ and $\beta _i$ are positive constant and $I_{ji}(t)$ is the inventory level for jth ($j=1, 2, 3, \ldots , k$) warehouse at time t.

• (ii) All warehouses are sequenced according to the order of the holding cost small to big.

• (iii) Rate of replenishment is infinite, and the replenishment size is finite.

• (iv) Deterioration start before kth warehouse inventory level reaches zero, i.e., $t^{d}_{i}<T_{1i}$ for ith item.

• (v) The model deals with multi-warehouse and multi-product of the retailer.

• (vi) There is no replacement or repair of deteriorating items during the period under consideration.

• (vii) Shortages are allowed. Unsatisfied demand is partially backlogged at a rate $\delta (t)$, which is a differentiable and non-increasing function of time t.

• (viii) Items in the kth warehouse are firstly used to meet the demand. Then, items in the $(k-1)$th warehouse are used, and so on.

3 Preliminaries

This section presented some basic concept and theorems on fuzzy rough theory. This results are important for remaining of the this paper.

Definition 3.1

(Rough space Xu and Zhou 2009) Let $\Lambda$ be a non-empty set, ${\mathscr {A}}$ a $\sigma$ algebra of subsets of $\Lambda$, and $\Delta$ an element in ${\mathscr {A}}$ and $\pi$ a trust measure, Then, $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ is called a rough space.

Definition 3.2

(Rough variable Xu and Zhou 2009) Let $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ be rough space. A rough variable $\zeta$ is a measurable function from the rough space $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ to the set of real numbers ${\mathbb {R}}$. That is, for every Borel set ${\mathbb {B}}$ of ${\mathbb {R}}$, we have

$$\begin{aligned} \{\eta \in \Lambda : \zeta (\eta )\in {\mathbb {B}}\} \in {\mathscr {A}} \end{aligned}$$

The upper $(\overline{\zeta })$ and lower $(\underline{\zeta })$ approximations of the rough variable $\zeta$ are defined as follows:

$$\begin{aligned} \overline{\zeta }=\{\zeta (\eta ) : \eta \in \Lambda \}~~~~~~ \underline{\zeta }=\{\zeta (\eta ) : \eta \in \Delta \} \end{aligned}$$

Definition 3.3

(Trust measure Xu and Zhao 2008) Let $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ be a rough space. The trust measure of the event A is defined by

$$\begin{aligned} \text {Tr}\{A\}=\dfrac{1}{2}(\underline{\text {Tr}}\{A\} + \overline{\text {Tr}}\{A\}) \end{aligned}$$

where the upper trust measure $\overline{\text {Tr}}\{A\} = \dfrac{\pi \{A\}}{\pi \{\Lambda \}}$ and lower trust measure $\underline{\text {Tr}}\{A\} = \dfrac{\pi \{A \cap \Delta \}}{\pi \{\Delta \}}$. When the enough information about the measure $\pi$ is not given. For this case, the measure $\pi$ may be treated as the Lebesgue measure.

Example 3.1

Let $\zeta =([a_{1}, a_{2}] [b_{1}, b_{2}])$ be a rough variable with $b_{1} \le a_{1} \le a_{2} \le b_{2}$ representing the identity function $\zeta (\eta )= \eta$ from the rough space $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ to the set of real numbers ${\mathbb {R}}$, where $\Lambda = \{\eta : b_{1} \le \eta \le b_{2}\}$, $\Delta = \{\eta : a_{1} \le \eta \le a_{2} \}$, ${\mathscr {A}}$ is the $\sigma$-algebra on $\Lambda$, and $\pi$ is the Lebesgue measure.

According to Definitions 3.2 and 3.3, we can obtain the trust measure of the event $\{\zeta \ge t\}$ and $\{\zeta \le t\}$ as follows:

$$\begin{aligned} \text {Tr}(\zeta \ge t)=\left\{ \begin{array}{ll}0 &{} \quad \hbox { if}\ b_{2} \le t\\ \dfrac{b_{2}-t}{2(b_{2}-b_{1})} &{} \quad \hbox { if}\ a_{2} \le t \le b_{2} \\ \dfrac{1}{2} \left( \dfrac{b_{2}-t}{b_{2}-b_{1}}+\dfrac{a_{2}-t}{a_{2}-a_{1}}\right) &{} \quad \hbox { if}\ a_{1} \le t \le a_{2} \\ \dfrac{1}{2}\left( \dfrac{b_{2}-t}{b_{2}-b_{1}} + 1\right) &{} \quad \hbox { if}\ b_{1} \le t \le a_{1}\\ 1 &{} \quad \hbox { if}\ t \le b_{1} \end{array}\right. \end{aligned}$$

(1)

$$\begin{aligned} \text {Tr}(\zeta \le t)=\left\{ \begin{array}{ll}0 &{} \quad \hbox { if}\ t \le b_{1}\\ \dfrac{t-b_{1}}{2(b_{2}-b_{1})} &{} \quad \hbox { if}\ b_{1} \le t \le a_{1} \\ \dfrac{1}{2} \left( \dfrac{t-b_{1}}{b_{2}-b_{1}}+\dfrac{t-a_{1}}{a_{2}-a_{1}}\right) &{} \quad \hbox { if}\ a_{1} \le t \le a_{2} \\ \dfrac{1}{2}\left( \dfrac{t -b_{1}}{b_{2}-b_{1}} + 1\right) &{} \quad \hbox { if}\ a_{2} \le t \le b_{2}\\ 1 &{} \quad \hbox { if}\ b_{2} \le t \end{array}\right. \nonumber \\ \end{aligned}$$

(2)

Example 3.2

Let us consider the triangular fuzzy variable $\zeta$ with the following membership function

$$\begin{aligned} \mu _{\zeta }(t) = \left\{ \begin{array}{ll} \dfrac{t-r_{1}}{r_{2}-r_{1}} &{} \quad \hbox { if}\ r_{1}\le t<r_{2}\\ 1 &{} \quad \hbox { if}\ t = r_{2}\\ \dfrac{r_{3}-t}{r_{3}-r_{2}} &{} \quad \hbox { if}\ r_{2}< t \le r_{3}\\ 0 &{} \quad \text{ otherwise } \end{array}\right. \end{aligned}$$

(3)

where every $r_{i}$ is a positive real number for $i=1,2,3$. Now, we assume that every $r_{i} \vdash ([a_{1}, a_{2}] [b_{1}, b_{2}])$ is rough variable for $i=1, 2, 3$. Then, $\zeta$ is called triangular fuzzy rough variable.

Definition 3.4

(Xu and Zhao 2010) A fuzzy rough variable is a measurable function from a rough space to $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ to the set of fuzzy variables such that $\text {Pos}\{\zeta (\eta ) \in {\mathbb {B}}\}$ is a measurable function of $\eta$ for any Borel set ${\mathbb {B}}$ of ${\mathbb {R}}$. Usually, say that a fuzzy rough variable is a rough variable taking fuzzy values.

Definition 3.5

(Xu and Zhao 2010) An n-dimensional fuzzy rough vector is a function $\zeta$ from a rough space $(\Lambda , \Delta , {\mathscr {A}}, \pi )$ to the set of n-dimensional fuzzy vectors such that $\text {Pos}\{\zeta (\eta ) \in {\mathbb {B}}\}$ is a measurable function of $\zeta$ for any Borel set ${\mathbb {B}}$ of ${\mathbb {R}}^{n}$.

Definition 3.6

(Xu and Zhao 2010) Let $f: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}$ be a function, and $\zeta _{1}, \zeta _{2}, \ldots , \zeta _{n}$ are fuzzy variables defined on $(\Lambda , \Delta , {\mathscr {A}}, \pi )$, respectively. Then, $\zeta =f(\zeta _{1}, \zeta _{2}, \ldots , \zeta _{n})$ is a fuzzy rough variable defined as $\zeta (\eta )=f(\zeta _{1}(\eta ), \zeta _{2}(\eta ), \ldots , \zeta _{n}(\eta ))$, for any $\eta \in \Lambda$

Definition 3.7

(Xu and Zhao 2010) Let $\zeta = (\zeta _{1}, \zeta _{2}, \ldots , \zeta _{n})$ be a fuzzy rough vector on the rough space $(\Lambda , \Delta , {\mathscr {A}}, \pi )$, and $g_{j}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}$ be continuous functions, $j=1,2,\ldots ,q$. Then, the primitive chance of a fuzzy event characterized by $g_{j}(\zeta ) \le 0, j=1,2, \ldots , q$ is a function from [0, 1] to [0, 1], defined as

$$\begin{aligned} Ch\{g_{j}(\zeta )\le & {} 0, j=1, 2, \ldots , q\}(\alpha )\\= & {} \sup \{\beta | \text {Tr} \{\eta \in \Lambda | \text {Pos} \{g_{j}(\zeta (\eta )) \\\le & {} 0, j=1, 2, \ldots , q\}\ge \beta \}\ge \alpha \} \end{aligned}$$

Proposition 3.1

(Xu and Zhao 2010) Let $\zeta$ be a fuzzy rough vector, i.e., with the n-tuple of fuzzy rough variables $(\zeta _{1}, \zeta _{2}, \ldots , \zeta _{n})$, and $g_{j}$ are real valued continuous functions for $j=1, 2, \ldots , q$. Then, the possibility $\text {Pos}\{g_{j}(\zeta (\eta )) \le 0, j=1, 2, \ldots , q\}$ is a rough variable.

Theorem 1

(Xu and Zhao 2010) Assume that $\widehat{c}_{ij}$ is a fuzzy rough variable, for any $\lambda \in \Lambda$, the fuzzy variable $\widehat{c}_{ij}(\lambda )$ is characterized by the following membership function

where $\gamma ^{c}_{ij}, \delta ^{c}_{ij}$ are positive numbers expressing the left and right spread of $\widehat{c}_{ij}(\lambda )$, reference functions $L, R: [0, 1] \rightarrow [0, 1]$ with $L(1)=R(1)=0$, and $L(0)=R(0)=1$ are non-increasing, continuous functions. And $(c_{ij}(\lambda )_{n \times 1}=(c_{i1}(\lambda ), c_{i2}(\lambda ), \ldots , c_{in}(\lambda ))^{T}$ is a rough vector. It follows that $c_{i}(\lambda )^{T}x=([a,b][c,d])$, where $c\le a<b\le d)$ is a rough variable and characterized by the following trust measure function

$$\begin{aligned} \text {Tr}\{c_{i}(\lambda )^{T}x \ge t\}=\left\{ \begin{array}{ll}0 &{} \quad \text { if}\ d\le t\\ \frac{d-t}{2(d-c)} &{} \quad \text { if}\ b \le t \le d \\ \frac{1}{2} \left( \frac{d-t}{d-c}+\frac{b-t}{b-a}\right) &{}\quad \text { if}\ a \le t < b \\ \frac{1}{2}\left( \frac{d-t}{d-c} + 1\right) &{} \quad \text { if}\ c \le t \le a\\ 1 &{}\quad \hbox { if}\ t \le c \end{array}\right. \end{aligned}$$

(4)

Then, we have $\text {Tr}\{\lambda | \text {Pos}\{\widehat{c}_{i}(\lambda )^{T} \ge f_{i}\}\ge \beta _{i}\} \ge \alpha _{i}$, if and only if

$$\begin{aligned} \left\{ \begin{array}{ll} f_{i} \le d-2\alpha _{i}(d-c)+R^{-1}(\beta _i)\delta _{i}^{cT}x &{}\quad \hbox { if}\ b \le f_{i} -R^{-1}(\beta _{i})\delta _{i}^{cT}x \le d \\ f_{i} \le \frac{d(b-a)+b(d-c)-2\alpha _{i}(d-c)(b-a)}{d-c+b-a}+R^{-1}(\beta _{i})\delta _{i}^{cT}x &{} \quad \hbox { if}\ a \le f_{i} -R^{-1}(\beta _{i})\delta _{i}^{cT}x < b \\ f_{i} \le d-(d-c)(2\alpha _{i}-1)+R^{-1}(\beta _{i})\delta _{i}^{cT}x &{}\quad \hbox { if}\ c \le f_{i}-R^{-1}(\beta _{i})\delta _{i}^{cT}x \le a\\ f_{i} \le c+ R^{-1}(\beta _{i})\delta _{i}^{cT}x &{} \quad \hbox {if } f_{i} - R^{-1}(\beta _{i})\delta _{i}^{cT}x \le c \end{array}\right. \end{aligned}$$

(5)

where $\alpha _{i}, \beta _{i} \in [0,1]$ are predetermined confidence levels.

Theorem 2

(Xu and Zhao 2010) Assume that $\widehat{e}_{kj}$, $\widehat{b}_{k}$ are fuzzy rough variable, for any $\lambda \in \Lambda$, fuzzy variables $\widehat{e}_{kj}(\lambda )$, $\widehat{b}_{k}(\lambda )$ are characterized by the membership function in the following

$$\begin{aligned} \mu _{\widehat{e}_{kj}(\lambda )}(t) = \left\{ \begin{array}{ll} L\bigg (\frac{e_{kj}(\lambda )-t}{\gamma ^{e}_{kj}}\bigg ) &{} \quad \hbox { if}\ t \le e_{kj}(\lambda ), \gamma ^{e}_{kj}>0\\ R\bigg (\frac{t-e_{kj}(\lambda )}{\delta ^{e}_{kj}}\bigg ) &{} \quad \text{ if } t \ge e_{kj}(\lambda ), \delta ^{e}_{kj}>0 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \mu _{\widehat{b}_{k}(\lambda )}(t) = \left\{ \begin{array}{ll} L\bigg (\frac{b_{k}(\lambda )-t}{\gamma ^{b}_{k}}\bigg ) &{} \quad \hbox { if}\ t \le b_{k}(\lambda ), \gamma ^{b}_{k}>0\\ R\bigg (\frac{t-b_{k}(\lambda )}{\delta ^{b}_{k}}\bigg ) &{}\quad \text{ if } t \ge b_{k}(\lambda ), \delta ^{b}_{k }>0 \end{array}\right. \end{aligned}$$

where $\gamma ^{e}_{kj}, \delta ^{e}_{kj}$ are positive numbers expressing the left and right spread of $\widehat{e}_{kj}(\lambda )$, $\gamma ^{b}_{k}, \delta ^{b}_{k}$ are the left and right spread of $\widehat{b}_{k}(\lambda )$, reference functions $L, R: [0, 1] \rightarrow [0, 1]$ with $L(1)=R(1)=0$, and $L(0)=R(0)=1$ are non-increasing, continuous functions. And $(e_{kj}(\lambda )_{n \times 1}=(c_{k1}(\lambda ), c_{k2}(\lambda ), \ldots , c_{kn}(\lambda ))^{T}$ is a rough vector, $e_{kj}(\lambda ), b_{k}(\lambda )$ are rough variables, $k=1,2, \ldots , p$, $jj=1,2,\ldots , n$. By Theorem 1, we have $e_{k}(\lambda )^{T}x, b_{k}(\lambda )$ are rough variables, then $e_{k}(\lambda )^{T}x-b_{k}(\lambda )=([a, b], [c, d]), (c\le a< b\le d)$ is also a rough variable. We assume that it is characterized by the following trust measure function

$$\begin{aligned}&\text {Tr}\{e_{k}(\lambda )^{T}x -b_{k}(\lambda )\le t\}\nonumber \\&\quad =\left\{ \begin{array}{ll}0 &{}\quad \hbox { if}\ t\le c\\ \frac{t-c}{2(d-c)} &{} \quad \hbox { if}\ c \le t \le a \\ \frac{1}{2} \left( \frac{t-c}{d-c}+\frac{t-a}{b-a}\right) &{} \quad \hbox { if}\ a \le t < b \\ \frac{1}{2}\left( \frac{t-c}{d-c} + 1\right) &{} \quad \hbox { if}\ b \le t \le d\\ 1 &{}\quad \hbox { if}\ t \ge d \end{array}\right. \end{aligned}$$

(6)

Then, we have $\text {Tr}\{\lambda | \text {Pos}\{\widehat{e}_{k}(\lambda )^{T}x \le \widehat{b}_{k}(\lambda )\} \ge \sigma _{k}\} \ge \eta _{k}$, if and only if

$$\begin{aligned} \left\{ \begin{array}{ll} W \ge c+2(d-c)\eta _{k} &{} \quad \hbox { if}\ c\le W \le a \\ W \ge \frac{2\eta _{k}(d-c)(b-a)+c(b-a)+a(d-c)}{d-c+b-a} &{}\quad \hbox { if}\ a \le W < b \\ W \ge (2\eta _{k}-1)(d-c)+c &{} \quad \hbox { if}\ b \le W \le d\\ W\ge d &{} \quad \hbox { if}\ d \le W \end{array}\right. \end{aligned}$$

(7)

where $W=R^{-1}(\sigma _{k})\delta ^{b}_{k}+L^{-1}(\sigma _{k})\gamma ^{eT}_{k}x$ and $\sigma _{k}, \eta _{k} \in [0,1]$ are predetermined confidence levels.

4 Mathematical model formulation

In this section, multi-warehouse multi-item partial backlogging inventory model with stock-dependent demand rate for non-instantaneous deteriorating items under inflation is developed. Here, we have considered k warehouses out of which $(k-1)$ warehouses are rented and one is own warehouse. For convenience, k warehouses are sequenced again according to the order of the holding cost in descending order and first warehouse is own warehouse. For economic reasons, kth warehouse are firstly used, then $(k-1)$th warehouse used to meet the demand and so on. The behavior of the model over the whole cycle $[0, T_i]$ is shown graphically in Fig. 1.

Fig. 1

Multi-warehouse inventory system for ith item

4.1 kth warehouse’s inventory system

During the time interval $[0,t^{d}_i{i}]$, the stock in the kth warehouse is depleted only due to demand and in the interval $[t^{d}_{i}, T_i]$ due tom demand and deterioration for ith item. The inventory level $I_{ki}$ for ith item during this time period governed by the following differential equations:

$$\begin{aligned} \frac{\mathrm{d}I_{ki}}{\mathrm{d}t}= & {} -(\alpha _i+\beta _iI_{ki}),\quad 0\le t\le t^{d}_{i} \end{aligned}$$

(8)

$$\begin{aligned} \frac{\mathrm{d}I_{ki}}{\mathrm{d}t}= & {} -(\alpha _i+\beta _iI_{ki})-\theta _iI_{ki},\quad t^{d}_{i} \le t\le T_{1i} \end{aligned}$$

(9)

The solutions of the above two differential Eqs. (8) and (9) with boundary conditions $I_{ki}(0)=W_{ki}$,$I_{ki}(T_{1i})=0$ and $I_{ki}(t^{d^{+}}_{i})=I_{ki}(t^{d^{-}}_{i})$, respectively, are

$$\begin{aligned} I_{ki}(t)= & {} W_{ki}e^{-\beta _it}+\frac{\alpha _i}{\beta _i}(1-e^{-\beta _it}),\quad 0 \le t\le t^{d}_{i} \end{aligned}$$

(10)

$$\begin{aligned} I_{ki}(t)= & {} \frac{\alpha _i}{\beta _i+\theta _i}(e^{(beta_i+\theta _i)(T_{1i}-t)}-1),\quad t^{d}_i \le t\le T_{1i}\nonumber \\ \end{aligned}$$

(11)

Now from Eqs.  (10) and (11), we get

$$\begin{aligned} W_{ki}= & {} \frac{\alpha _i}{\beta _i+\theta _i}(e^{(beta_i+\theta _i)(T_{1i}-t^{d}_{i})}-1)e^{\beta _it^{d}_{i}}\nonumber \\&-\frac{\alpha _i}{\beta _i}(1-e^{-\beta _it^{d}_{i}}) \end{aligned}$$

(12)

4.2 $(k-j)$th warehouse’s inventory system

During the time interval $[0,t^{d}_i]$, the stock in the $(k-j)$th warehouse $(j=1, 2, \ldots , k-1)$, there is no change in inventory level, because during this period there is neither deterioration nor supply of items from $(k-j)$th warehouses. Thus at any time t, inventory level is

$$\begin{aligned} I_{(k-j)i}(t)=W_{(k-j)i}, \quad 0\le t\le t^{d}_{i} \end{aligned}$$

(13)

Afterward, the inventory depletes only due to deterioration up to time $t=T_{ji}$; the differential equation is

$$\begin{aligned} \frac{\mathrm{d}I_{(k-j)i}(t)}{\mathrm{d}t}=-\theta _iI_{(k-j)i}(t), ~~t^{d}_{i}\le t\le T_{ji} \end{aligned}$$

(14)

After time $t=T_{ji}$, demand is met from OW, and hence, inventory level decreases due to both demand and deterioration during the interval $[T_{ji}, T_{(j+1)i}]$ and falls to zero at $t=T_{(j+1)i}$; the differential equation is

$$\begin{aligned} \frac{\mathrm{d}I_{(k-j)i}(t)}{\mathrm{d}t}= & {} -(\alpha _i+\beta _iI_{(k-j)i})\nonumber \\&-\theta _iI_{(k-j)i}(t), ~~T_{ji}\le t\le T_{(j+1)i} \end{aligned}$$

(15)

Now there after during the time interval $[T_{ki}, T_i]$, the demand is partially backlogged. The backlogged level, $B_i(t)$ during the interval $[T_{ki}, T_i]$, satisfies the following differential equation:

$$\begin{aligned} \frac{\mathrm{d}B_i(t)}{\mathrm{d}t}=\alpha _i\delta _i(T_i-t), ~~T_{ki}\le t\le T_i \end{aligned}$$

(16)

The solutions of the above two differential equations (14), (15) and (16) with boundary conditions $I_{(k-j)i}(0)=W_{(k-j)i}$, $I_{(k-j)i}(T_{(j+1)i})=0$, $B_i(T_{ki})=0$ and $I_{(k-j)i}(t^{d^{+}}_{i})=I_{(k-j)i}(t^{d^{-}}_{i})$, respectively, are

$$\begin{aligned} I_{(k-j)i}(t)= & {} W_{(k-j)i}e^{\theta _i(t^{d}_{i}-t)},\quad t^{d}_{i} \le t\le T_{ji} \end{aligned}$$

(17)

$$\begin{aligned} I_{(k-j)i}(t)= & {} \frac{\alpha _i}{\beta _i+\theta _i}(e^{(\beta _i+\theta _i)(T_{(j+1)i}-t)}-1), \nonumber \\ T_{ji}\le & {} t\le T_{(j+1)i} \end{aligned}$$

(18)

$$\begin{aligned} B_{i}(t)= & {} \frac{\alpha _i}{\sigma _i}e^{-\sigma _iT_i}(e^{\sigma _it}-e^{\sigma _iT_{ki}})\quad T_{ki} \le t\le T_{i} \end{aligned}$$

(19)

Now from Eqs. (17) and (18), we get

$$\begin{aligned} W_{(k-j)i}=\frac{\alpha _i}{\beta _i+\theta _i}(e^{(\beta _i+\theta _i)(T_{(j+1)i}-T_{ji})}-1)e^{-\theta _i(t^{d}_{i}-T_{ji})}\nonumber \\ \end{aligned}$$

(20)

4.3 Retailer’s total profit

Thus, the present worth of the total profit per cycle for the inventory system consists of the following components: the cumulative inventory during $(0,T_{1i})$ in kth warehouse for ith item under inflation is $\int \nolimits _{0}^{T_{1i}}I_{ki}(t)e^{-Rt} \mathrm{d}t$ and during $(0, T_{(j+1)i})$ in $(k-j)$th warehouse for ith item under inflation is $\int \nolimits _{0}^{T_{(j+1)i}}I_{(k-j)i}(t)e^{-Rt} \mathrm{d}t$. As a result, the present value of the inventory holding cost for ith item is

$$\begin{aligned} \text {HC}_i= & {} H_k\bigg [\int \limits _{0}^{T_{1i}}I_{ki}(t)e^{-Rt}\mathrm{d}t \nonumber \\&+\sum \limits _{j=1}^{k-1}H_{k-j}\int \limits _{0}^{T_{(j+1)i}}I_{(k-j)i}(t)e^{-Rt}\mathrm{d}t\bigg ]\nonumber \\= & {} H_{k}\bigg [\frac{W_{ki}}{R+\beta _i}(1-e^{-(R+\beta _i)t^{d}_{i}})+\frac{\alpha _{i}}{\beta _iR}(e^{-Rt^{d}_{i}}-1)\nonumber \\&+\frac{\alpha _i}{\beta _i(\beta _i+R)}(1-e^{-(\beta _i+R)t^{d}_{i}})\nonumber \\&+\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{1i}}}{\beta _i+\theta _i+R}\nonumber \\&\quad \bigg (e^{-(\beta _i+\theta _i+R)t^{d}_{i}}-e^{-(\beta _i+\theta _i+R)T_{1i}}\bigg )\nonumber \\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{1i}}-e^{-Rt^{d}_{i}}\bigg )\bigg ]\nonumber \\&+\sum \limits _{j=1}^{k-1}H_{k-j}\bigg [\frac{W_{(k-j)i}}{R}(1-e^{-Rt^{d}_{i}})\nonumber \\&+\frac{W_{(k-j)i}e^{\theta _it^{d}_{i}}}{R+\theta _i}\bigg (e^{-(R+\theta _i)t^{d}_{i}}-e^{-(R+\theta _i)T_{ji}}\bigg )\nonumber \\&+\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{(j+1)i}}}{\beta _i+\theta _i+R}\nonumber \\&\quad \bigg (e^{-(\beta _i+\theta _i+R)T_{ji}}-e^{-(\beta _i+\theta _i+R)T_{(j+1)i}}\bigg )\nonumber \\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{(j+1)i}}-e^{-RT_{ji}}\bigg )\bigg ] \end{aligned}$$

(21)

The present value of the deterioration cost for ith item is

$$\begin{aligned} \text {DC}_i= & {} p_i\theta _i\bigg [\int \limits _{t^{d}_{i}}^{T_{1i}}I_{ki}(t)e^{-Rt}\mathrm{d}t+\sum \limits _{j=1}^{k-1}\int \limits _{t^{d}_{i}}^{T_{(j+1)i}}I_{(k-j)i}(t)e^{-Rt}\mathrm{d}t\bigg ]\nonumber \\&\quad \theta _i\bigg [\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{1i}}}{\beta _i+\theta _i+R}\bigg (e^{-(\beta _i+\theta _i+R)t^{d}_{i}}-e^{-(\beta _i+\theta _i+R)T_{1i}}\bigg )\nonumber \\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{1i}}-e^{-Rt^{d}_{i}}\bigg )\nonumber \\&+\sum \limits _{j=1}^{k-1}\bigg \{\frac{W_{(k-j)i}e^{\theta _it^{d}_{i}}}{R+\theta _i}\bigg (e^{-(R+\theta _i)t^{d}_{i}}\nonumber \\&-e^{-(R+\theta _i)T_{ji}}\bigg )+\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{(j+1)i}}}{\beta _i+\theta _i+R}\nonumber \\&\quad \bigg (e^{-(\beta _i+\theta _i+R)T_{ji}}-e^{-(\beta _i+\theta _i+R)T_{(j+1)i}}\bigg )\nonumber \\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{(j+1)i}}-e^{-RT_{ji}}\bigg )\bigg \}\bigg ] \end{aligned}$$

(22)

The present value of the backlogging cost and the opportunity cost due to lost sale for ith item are

$$\begin{aligned} \text {BC}_i= & {} c^{b}_{i}\int \limits _{T_{ki}}^{T_{i}}B_{i}(t)e^{-Rt} \mathrm{d}t\nonumber \\= & {} \frac{\alpha _i e^{-\sigma _iT_i}c^{b}_{i}}{\sigma _i}\bigg [\frac{1}{\sigma _i-R}(e^{(\sigma _i-R)T_i}-e^{(\sigma _i-R)T_{ki}})\nonumber \\&+\frac{e^{\sigma _iT_{ki}}}{R}(e^{-RT_i}-e^{-RT_{ki}})\bigg ] \end{aligned}$$

(23)

and

$$\begin{aligned} \text {OCL}_i= & {} c^{l}_{i}\alpha _i\int \limits _{T_{ki}}^{T_{i}}(1-e^{-\sigma _i(T_i-t)}) e^{-Rt}\mathrm{d}t\nonumber \\= & {} \alpha _i c^{l}_{i}\bigg [\frac{1}{R}(e^{-RT_{ki}}-e^{-RT_i})\nonumber \\&+\frac{e^{-\sigma _iT_i}}{R-\sigma _i}(e^{-(R-\sigma _i)T_i}-e^{-(R-\sigma )T_{ki}})\bigg ] \end{aligned}$$

(24)

The present worth of the revenue for ith item is

$$\begin{aligned} \text {SR}_i= & {} s_{i}\bigg [\int \limits _{0}^{T_{1i}}D_i(t)e^{-Rt} \mathrm{d}t+ \int \limits _{0}^{T_{(j+1)i}}D_i(t)e^{-Rt} \mathrm{d}t \nonumber \\&+\frac{\alpha _ie^{-RT_i}}{\sigma _i}\bigg (1-e^{\sigma _i(T_{ki}-T_i)}\bigg )\bigg ]\nonumber \\= & {} s_i\bigg [ \frac{\alpha _i}{R}(1-e^{-RT_{1i}})+\frac{\beta _iW_{ki}}{\beta _i+R}(1-e^{-(\beta _i+R)t^{d}_{i}})\nonumber \\&+\frac{\alpha _i}{R}(e^{-Rt^{d}_{i}}-1)\nonumber \\&+\frac{\alpha _i}{\beta _i+R}(e^{-(\beta _i+R)t^{d}_{i}}-1){+} \frac{\alpha _i\beta _i e^{(\beta _i+\theta _i)T_{1i}}}{(\beta _i+\theta _i)(\beta _i+\theta _i+R)}\nonumber \\&\quad \bigg (e^{-(\beta _i+\theta _i+R)t^{d}_{i}}-e^{-(\beta _i+\theta _i+R)T_{1i}}\bigg )\nonumber \\&+\frac{\alpha _i\beta _i}{(\beta _i+\theta _i)R}(e^{-RT_{1i}}-e^{-Rt^{d}_{i}})\nonumber \\&+\sum \limits _{j=1}^{k-1}\bigg \{\frac{\alpha _i}{R}(1-e^{-RT_{(j+1)i}})+\frac{\beta _iW_{(k-j)i}}{R}(1-e^{-Rt^{d}_{i}})\nonumber \\&+\frac{\beta _iW_{(k-j)i}e^{\theta _it^{d}_i}}{R+\theta _i}(e^{-(R+\theta _i)t^{d}_{i}}-e^{-(R+\theta _i)T_{ji}}) \nonumber \\&\quad \frac{\alpha _i\beta _i e^{(\beta _i+\theta _i)T_{(j+1)i}}}{(\beta _i+\theta _i)(\beta _i+\theta _i+R)}\nonumber \\&\quad \bigg (e^{-(\beta _i+\theta _i+R)T_{ji}}-e^{-(\beta _i+\theta _i+R)T_{(j+1)i}}\bigg )\nonumber \\&+\frac{\alpha _i\beta _i}{(\beta _i+\theta _i)R}(e^{-RT_{(j+1)i}}-e^{-RT_{ji}})\bigg \}\nonumber \\&+\frac{\alpha _ie^{-RT_i}}{\sigma _i}\bigg (1-e^{\sigma _i(T_{ki}-T_i)}\bigg )\bigg ] \end{aligned}$$

(25)

The total units ($\text {TT}^{k-j+1}_{i}$) of ith item transferred from $(k-j+1)$th $(j=1,2,3,\ldots ,k-1)$ to 1st warehouse are

$$\begin{aligned} \text {TT}^{k-j+1}_{i}= & {} W_{(k-j+1)i}\\&-\int \limits _{t^{d}_{i}}^{T_{ji}}\theta _i I_{(k-j+1)i}(t)\mathrm{d}t\quad j=1,2,3,\ldots ,(k-1) \end{aligned}$$

The transportation cost ($\text {TC}^{k-j+1}_i$) for transporting $\text {TT}^{k-j+1}_{i}$ units from $(k-j+1)$th $\bigg (j=1,2,3,\ldots ,(k-1)\bigg )$ (Rented warehouse) to 1st (Owned warehouse) warehouse is

$$\begin{aligned} \text {TC}^{k-j+1}_i= & {} \text {TC}^{1}_{i}+\text {TT}^{k-j+1}_{i}\times d_{k-j+1} \\&\times \text {TCP}^{k-j+1}_{i},\quad j=1,2,3,\ldots ,(k-1) \end{aligned}$$

The present value of the total transportation cost for ith item is

$$\begin{aligned} \text {TC}_i=\sum \limits _{j=1}^{k-1}\text {TC}^{k-j+1}_{i} \end{aligned}$$

(26)

The present worth of the purchasing cost for ith item is

$$\begin{aligned} \text {PC}_i= & {} (W_{1i}+W_{2i}+W_{3i})p_i+\frac{p_ie^{-RT_i}}{\sigma _i}\bigg (1-e^{\sigma _i(T_{ki}-T_i)}\bigg )\nonumber \\ \end{aligned}$$

(27)

Therefore, the retailer’s average total profit ($\text {TP}$) is given by

$$\begin{aligned} \text {TP}= & {} \sum \limits _{i=1}^{m}\text {TP}_i =\sum \limits _{i=1}^{m}\bigg (\text {SR}_i-\text {PC}_i-\text {SC}_i-\text {HC}_i\nonumber \\&-\text {DC}_i-\text {TC}_i-\text {BC}_i-\text {OCL}_i\bigg )/T_i \end{aligned}$$

(28)

5 Multi-warehouse multi-item inventory model under fuzzy rough environment

In real-life situation, various parameters like ordering cost, purchasing cost, selling price, holding cost, transportation cost, revenue cost, backlogging cost and opportunity cost of multi-warehouse multi-item inventory model are uncertain in nature. In this section, multi-warehouse multi-item inventory model is considered by using up those cost in fuzzy rough environments.

$$\begin{aligned} \text{ Maximize }&\widehat{\widetilde{\text {TP}}}\nonumber \\ \text{ subject } \text{ to }&\sum \limits _{i=1}^{m}\widehat{\widetilde{\text {TCR}}}_i\le B\\&\sum \limits _{i=1}^{m} A_iQ_i\le A \nonumber \end{aligned}$$

(29)

Table 2 Input data in deterministic environments

Table 3 Parameter value of fuzzy rough variables

Table 4 The optimal result of the multi-warehouse multi-item problem

Table 5 Optimal result for different inflation rate

Table 6 Optimal result for different demand rate

Table 7 Optimal result for different backlogging rate

Table 8 Optimal result for different deterioration rate

Table 9 Impact of non-deteriorating period($t^d_i$) on the optimal replenishment policy

where

$$\begin{aligned} \widehat{\widetilde{\text {TP}}}= & {} \sum \limits _{i=1}^{m}\widehat{\widetilde{\text {TP}}}_i\nonumber \\= & {} \sum \limits _{i=1}^{m}\bigg (\widehat{\widetilde{\text {SR}}}_i-\widehat{\widetilde{\text {PC}}}_i-\widehat{\widetilde{\text {SC}}}_i-\widehat{\widetilde{\text {HC}}}_i \nonumber \\&-\widehat{\widetilde{\text {DC}}}_i-\widehat{\widetilde{\text {TC}}}_i-\widehat{\widetilde{\text {BC}}}_i-\widehat{\widetilde{\text {OCL}}}_i\bigg )/T_i \end{aligned}$$

(30)

$$\begin{aligned} \widehat{\widetilde{\text {TCR}}}_i= & {} \bigg (\widehat{\widetilde{p}}_iQ_i+\widehat{\widetilde{\text {SC}}}_{i}+\widehat{\widetilde{\text {HC}}}_i\bigg )/T_i \end{aligned}$$

(31)

$$\begin{aligned} \widehat{\widetilde{\text {SR}}}_i= & {} \widehat{\widetilde{s}}_i\bigg [ \frac{\alpha _i}{R}(1-e^{-RT_{1i}})+\frac{\beta _iW_{ki}}{\beta _i+R}(1-e^{-(\beta _i+R)t^{d}_{i}})\\&+\frac{\alpha _i}{R}(e^{-Rt^{d}_{i}}-1)+\frac{\alpha _i}{\beta _i+R}(e^{-(\beta _i+R)t^{d}_{i}}-1)\\&+ \frac{\alpha _i\beta _i e^{(\beta _i+\theta _i)T_{1i}}}{(\beta _i+\theta _i)(\beta _i+\theta _i+R)}\\&\quad \bigg (e^{-(\beta _i+\theta _i+R)t^{d}_{i}}-e^{-(\beta _i+\theta _i+R)T_{1i}}\bigg )\\&+\frac{\alpha _i\beta _i}{(\beta _i+\theta _i)R}(e^{-RT_{1i}}-e^{-Rt^{d}_{i}})\\&+\sum \limits _{j=1}^{k-1}\bigg \{\frac{\alpha _i}{R}(1-e^{-RT_{(j+1)i}})\\&+\frac{\beta _iW_{(k-j)i}}{R}(1-e^{-Rt^{d}_{i}})\\&+\frac{\beta _iW_{(k-j)i}e^{\theta _it^{d}_i}}{R+\theta _i}(e^{-(R+\theta _i)t^{d}_{i}}-e^{-(R+\theta _i)T_{ji}}) \\&\quad \frac{\alpha _i\beta _i e^{(\beta _i+\theta _i)T_{(j+1)i}}}{(\beta _i+\theta _i)(\beta _i+\theta _i+R)}\\&\quad \bigg (e^{-(\beta _i+\theta _i+R)T_{ji}}-e^{-(\beta _i+\theta _i+R)T_{(j+1)i}}\bigg )\\&+\frac{\alpha _i\beta _i}{(\beta _i+\theta _i)R}(e^{-RT_{(j+1)i}}-e^{-RT_{ji}})\bigg \}\\&+\frac{\alpha _ie^{-RT_i}}{\sigma _i}\bigg (1-e^{\sigma _i(T_{ki}-T_i)}\bigg )\bigg ]\\ \widehat{\widetilde{\text {PC}}}_i= & {} (W_{1i}+W_{2i}+W_{3i})\widehat{p}_i\\&+\frac{\widehat{p}_ie^{-RT_i}}{\sigma _i}\bigg (1-e^{\sigma _i(T_{ki}-T_i)}\bigg )\\ \widehat{\widetilde{\text {DC}}}_i= & {} \widehat{\widetilde{p}}_i\theta _i\bigg [\int \limits _{t^{d}_{i}}^{T_{1i}}I_{ki}(t)e^{-Rt}\mathrm{d}t\\&+\sum \limits _{j=1}^{k-1}\int \limits _{t^{d}_{i}}^{T_{(j+1)i}}I_{(k-j)i}(t)e^{-Rt}\mathrm{d}t\bigg ]\\&\quad \theta _i\bigg [\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{1i}}}{\beta _i+\theta _i+R}\\&\quad \bigg (e^{-(\beta _i+\theta _i+R)t^{d}_{i}}-e^{-(\beta _i+\theta _i+R)T_{1i}}\bigg )\\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{1i}}-e^{-Rt^{d}_{i}}\bigg )\\&+\sum \limits _{j=1}^{k-1}\bigg \{\frac{W_{(k-j)i}e^{\theta _it^{d}_{i}}}{R+\theta _i}\bigg (e^{-(R+\theta _i)t^{d}_{i}}\\&-e^{-(R+\theta _i)T_{ji}}\bigg )+\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{(j+1)i}}}{\beta _i+\theta _i+R}\\&\quad \bigg (e^{-(\beta _i+\theta _i+R)T_{ji}}-e^{-(\beta _i+\theta _i+R)T_{(j+1)i}}\bigg )\\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{(j+1)i}}-e^{-RT_{ji}}\bigg )\bigg \}\bigg ]\\ \widehat{\text {HC}}_i= & {} \widehat{H}_{k}\bigg [\frac{W_{ki}}{R+\beta _i}(1-e^{-(R+\beta _i)t^{d}_{i}})\\&+\frac{\alpha _{i}}{\beta _iR}(e^{-Rt^{d}_{i}}-1)+\frac{\alpha _i}{\beta _i(\beta _i+R)}(1-e^{-(\beta _i+R)t^{d}_{i}})\\&+\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{1i}}}{\beta _i+\theta _i+R}\\&\quad \bigg (e^{-(\beta _i+\theta _i+R)t^{d}_{i}}-e^{-(\beta _i+\theta _i+R)T_{1i}}\bigg )\\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{1i}}-e^{-Rt^{d}_{i}}\bigg )\bigg ]\\&+\sum \limits _{j=1}^{k-1}\widehat{H}_{k-j}\bigg [\frac{W_{(k-j)i}}{R}(1-e^{-Rt^{d}_{i}})\\&+\frac{W_{(k-j)i}e^{\theta _it^{d}_{i}}}{R+\theta _i}\bigg (e^{-(R+\theta _i)t^{d}_{i}}-e^{-(R+\theta _i)T_{ji}}\bigg )\\&+\frac{\alpha _i}{\beta _i+\theta _i}\frac{e^{(\beta _i+\theta _i)T_{(j+1)i}}}{\beta _i+\theta _i+R}\\&\quad \bigg (e^{-(\beta _i+\theta _i+R)T_{ji}}-e^{-(\beta _i+\theta _i+R)T_{(j+1)i}}\bigg )\\&+\frac{\alpha _i}{(\beta _i+\theta _i)R}\bigg (e^{-RT_{(j+1)i}}-e^{-RT_{ji}}\bigg )\bigg ]\\ \widehat{\widetilde{\text {BC}}}_i= & {} \frac{\alpha _i e^{-\sigma _iT_i}\widehat{c}^{b}_{i}}{\sigma _i}\bigg [\frac{1}{\sigma _i-R}(e^{(\sigma _i-R)T_i}-e^{(\sigma _i-R)T_{ki}})\\&+\frac{e^{\sigma _iT_{ki}}}{R}(e^{-RT_i}-e^{-RT_{ki}})\bigg ]\\ \widehat{\widetilde{\text {TC}}}_i= & {} \sum \limits _{j=1}^{k-1}\widehat{\text {TC}}^{k-j+1}_{i}\\ \widehat{\widetilde{\text {TC}}}^{k-j+1}_i= & {} \widehat{\text {TC}}^{1}_{i}+\text {TT}^{k-j+1}_{i}\times d_{k-j+1} \\&\times \widehat{\text {TCP}}^{k-j+1}_{i},\quad j=1,2,3,\ldots ,(k-1) \\ \widehat{\widetilde{\text {OCL}}}_i= & {} \alpha _i \widehat{\widetilde{c}}^{l}_{i}\bigg [\frac{1}{R}(e^{-RT_{ki}}-e^{-RT_i})\\&+\frac{e^{-\sigma _iT_i}}{R-\sigma _i}(e^{-(R-\sigma _i)T_i}-e^{-(R-\sigma )T_{ki}})\bigg ]\\ \end{aligned}$$

and $\widehat{\widetilde{a}}$ represents that the parameter a is in fuzzy rough environment.

6 Equivalent deterministic representation of the proposed MWMIFR inventory model

To solve the proposed MWMIFR inventory model, we transform problem (29) into fuzzy rough Tr–Pos constrained inventory model using Theorems 1 and 2. Thus, we have the following fuzzy rough Tr–Pos constrained inventory model:

$$\begin{aligned} \begin{array}{ll} \text {max }f &{}\\ \text {s.t.}&{}\\ &{}\text {Tr}\{\lambda : \text {Pos}\{\widehat{\text {TP}}(\lambda )\ge f\}\ge \beta \}\ge \alpha \\ &{}\text {Tr}\{\lambda : \text {Pos}\{\sum \limits _{i=1}^{m}\widehat{\text {TCR}_{i}}(\lambda )\le B\}\ge \sigma \}\ge \eta \\ &{}\sum \limits _{i=1}^{n} A_{i}Q_{i}\le A\\ &{} T_{i}>0, i=1,2,\ldots ,n \end{array} \end{aligned}$$

Assume that the fuzzy variables $\widehat{\text {TP}}(\lambda )$ and $\sum \nolimits _{i=1}^{m}\widehat{\text {TCR}}_{i}$ are characterized by the membership function in the following:

$$\begin{aligned} \mu _{\widehat{\text {TP}}(\lambda )}(t) = \left\{ \begin{array}{ll} L\bigg (\frac{\text {TP}(\lambda )-t}{\gamma _{1}}\bigg ) &{} \hbox { if}\ t \le \text {TP}(\lambda ), \gamma _1>0\\ R\bigg (\frac{t-\text {TP}(\lambda )}{\delta _{1}}\bigg ) &{} \hbox { if }t \ge \text {TP}(\lambda ), \delta _1>0 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned}&\mu _{\sum \nolimits _{i=1}^{m}\widehat{\text {TCR}}_{i}(\lambda )}(t) \\&\quad = \left\{ \begin{array}{ll} L\bigg (\frac{\sum \nolimits _{i=1}^{m}\text {TCR}_{i}(\lambda )-t}{\gamma _{2}}\bigg ) &{} \hbox { if}\ t \le \sum \limits _{i=1}^{m}\text {TCR}_{i}(\lambda ), \gamma _2>0\\ R\bigg (\frac{t-\sum \nolimits _{i=1}^{m}\text {TCR}_{i}(\lambda )}{\delta _{2}}\bigg ) &{} \hbox { if }t \ge \sum \limits _{i=1}^{m}\text {TCR}_{i}(\lambda ), \delta _2>0 \end{array}\right. \end{aligned}$$

where $\gamma _1, \gamma _2, \delta _1, \delta _2$ are, respectively, positive numbers expressing the left and right spread of $\widehat{\text {TP}}(\lambda )$ and $\sum \nolimits _{i=1}^{m}\widehat{\widetilde{\text {TCR}}}_{i}(\lambda )$. We assume that the rough variables $\widehat{\text {TP}}(\lambda )=( [a_1, b_1],[c_1, d_1])$ and $\sum \nolimits _{i=1}^{m}\text {TCR}_{i}(\lambda )=([a_2, b_2], [c_2, d_2])$. Here, we consider the case when $b_1\le f_{i}R^{-1}(\beta _1)\delta _1^{T} T \le d_1$ and $b_2\le B+L^{-1}(\delta _2)\gamma _2^{T} T\le d_2$; then according to Theorems 1 and 2, problem (32) can be converted to its equivalent model as follows:

$$\begin{aligned} \text {max }&d_1-2\alpha (d_1-c_1)+R^{-1}(\beta )\delta _1^{T}T \nonumber \\ \text {s.t.}&\nonumber \\&B+L^{-1}(\sigma )\gamma _2^{T}T\ge (2\eta -1)(d_2-c_2)+c_2\ \nonumber \\&\sum \limits _{i=1}^{n} A_{i}Q_{i}\le A \nonumber \\&T_i>0, \sigma , \eta \in [0,1]\nonumber \\&\alpha , \beta \in [0, 1] \end{aligned}$$

(32)

7 Numerical illustration

A retail market of India buy their products in a large quantity which is not sufficient to store in their own warehouse, and they have rented more than one warehouse nearby to store their excess product. As per recruitment, they bring their items in their retail store to sell their products. Due to fluctuation of market price, fuel price, items price, etc., several parameters involved in this model considered under fuzzy rough environment. Data during one complete cycle retailer’s commercializing are collected and given in Tables 2 and 3. Here, we have considered that the retailer’s buy three types of product and stored excess products in two different places other than own warehouse. We have solved the problem stated in (32) using a possible measure at $0.9\text {-Pos}$ and $0.9\text {-Tr}$ level by Mathematica-8.0 software. The optimal results are shown in Table 4. The impact of net discount inflation rate, deterioration parameter, demand parameter and backlogging parameter is discussed in Tables 5, 6, 7, 8 and 9. The results obtained in Tables 5, 6, 7, 8 and 9 are also depicted in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11.

8 Sensitivity analysis

In this section, we perform the sensitivity analysis in order to study the impact of backlogging parameter ($\sigma$), rate of inflation (R), demand parameter $\alpha _i$ & $\beta _i$, deterioration rate ($\theta _i$) and period for deterioration ($t^d_i$) on the optimal cycle length ($T_i$) and the retailers total profit ($\text {TP}_i$). The results are presented in Tables 5, 6, 7, 8 and 9, which are also reflected in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11, and interesting findings are summarized as follows:

Fig. 2

Effect of net inflation rate on profit

Fig. 3

Effect of net inflation rate on cycle length

Fig. 4

Effect of backlogging parameter on profit

Fig. 5

Effect of backlogging parameter on cycle length

Fig. 6

Effect of deterioration rate on profit

Fig. 7

Effect of non-deteriorating period on profit

Fig. 8

Effect of $\alpha _i$ on profit

Fig. 9

Effect of $\alpha _i$ on cycle length

Fig. 10

Effect of $\beta _i$ on profit

Fig. 11

Effect of $\beta _i$ on cycle length

• (i) It is clearly visible from Table 7 that, as the backlogging parameter ($\sigma$) increases, i.e., backlogging rate decreases, there is a reduction in total profit ($\text {TP}_i$) and optimal cycle length($T_i$) (cf. Figs. 4, 5). Because decreasing the backlogging rate refers to lesser backlogged demand, thereby decreasing the order quantity, but at the same time the initial inventory for the cycle increases, which increases the holding cost, ultimately reducing the total profit.

• (ii) When the net discount rate of inflation (R) increases, i.e., inflation rate decreases, decrease the retailer’s total profit ($\text {TP}_i$) (Table 5). Because as the rate of inflation decreases, all the inventory-related cost decreases, which reduce the total cost of the inventory at the same time it decrease the sales revenue also, eventually decreases the total profit (cf. Fig. 2).

• (iii) As demand parameter $\alpha _i$ for each item increases, having other demand parameter $\beta _i$ at a constant level, it is observed from Table 6 (cf. Figs. 8, 9) that the total profit ($\text {TP}_i$) of each item increases, whereas cycle length ($T_i$) decreases for the multi-warehouse multi-item inventory problem. Increasing the demand parameter means more demand for the product in the market, which increase the sales revenue and thereby increases the total profit.

• (iv) From Table 6, it is observed that profit ($\text {TP}_i$) of each item increases, whereas cycle length ($T_i$) decreases for the multi-warehouse multi-item inventory problem with the increase in demand parameter $\beta _i$, keeping the other demand parameter fixed ($\alpha _i$) for each item (cf. Figs. 10, 11).

• (v) Further, Table 8 illustrates that as the deterioration rate ($\theta _i$) increases, the cycle length when inventory level reaches zero for different warehouses ($T_{ji}$) and profit ($\text {TP}_i$) for each item decreases (cf. Fig. 6). Because the increase in the deterioration rate causes the raise in the deterioration cost, ultimately reduces the total profit.

• (vi) It is observed from Table 9 that, as the period for deterioration($t^d_i$) increases, the deterioration cost for items decreases as a result retailer’s gain larger profits ($\text {TP}_i$) for each item (cf. Fig. 7).

9 Managerial insights

On the basis of sensitivity analysis of the parameters, we obtain the following managerial phenomena:

• When deterioration rates ($\theta _i$) of items increase, the total cost increases, whereas the consumption time of owned warehouse and rented warehouses inventories decreases, as a result total profit decreases. So it is suggestive that the retailer should maintain OW and store as much as possible goods in RW, to reduce the deterioration rates. This will minimize the deterioration cost and increase the profit of the retailer.

• As large the length of non-deterioration time is, smaller is the replenishment cycle thereby increasing its associated total profit. If the period of deterioration is delayed then the deterioration cost is reduce, this will increase the total profit.

• High inflation rate increases retailers relevant inventory costs and as a result decrease the retailers total profit. For higher inflationary environment, it is suggested that retailer can reduce the order quantity to reduce the total cost and hence increase the profit.

• A increase in backlogging parameter $\sigma$, that is, decreases backlogging rate, reduce retailer’s total profit and optimal cycle length. Since decreasing in backlogging rate implies less of backlogged demand, which ultimately reduces the total profit.

• When the value of demand parameters increases, optimal cycle length decreases and total profit increases. Higher demand leads to an increase of sales revenue, which in turn increases the profit. Therefore, it is suitable for the retailer’s to sale the items as much as possible before product starts deteriorating to increase the total profit.

• In a decision-making process, we may face a hybrid uncertain environment where fuzziness and roughness exist at the same time. For example, the inventory-related costs like holding cost, setup cost, purchase cost, selling price, transportation cost, etc., depend on several factors such as limited storage space, fluctuation in market, labor wages, etc., which are uncertain in fuzzy rough sense. So it is suggested the retailer to consider those parameter in fuzzy rough environment to get better assessment about the multi-warehouse multi-item inventory model.

10 Conclusions

In many cases, it is found that some parameters involve in an inventory system may not be crisp and be somewhat imprecise in nature. For example, holding cost for an item is supposed to be dependent on the amount put in the storage which may be imprecise and range on an interval due to several factors such as limited storage space, fluctuation in market, etc. Similarly, the setup cost depends upon the total quantity to be produced in a scheduling period range on an interval. In these situations, fuzzy rough theory can be used for the formulation of inventory systems. So, in the present paper, we have formulated and solved a multi-warehouse multi-item partial backlogging inventory model for deteriorating items under fuzzy rough environment. In the proposed model, we have considered the deterioration as non-instantaneous type, under this assumption items do not deteriorate immediately and maintain fresh quality during a span of time. In this model, we have also incorporated a type of partial lost sale into an inventory system by assuming it to be a function of shortages already backlogged. The demand of items is assumed to be stock dependent, and deterioration of items is assumed to be constant over time. Till now, no multi-warehouse inventory model has been formulated in such an environment. Due to several uncertainties of the environment, here the cost and other parameters are assumed to be fuzzy rough variables. We have used Tr–Pos chance constrained technique to convert the proposed fuzzy rough model the equivalent deterministic inventory model. The study determines the retailer’s optimal replenishment policies that maximizes the total profit. A numerical example and sensitivity analysis on key parameters has been provided to direct the retailer to take proper action under the different situation.

The proposed model can be further extended in several ways. We can extended the proposed single-objective problem to a multi-objective model. In the proposed model, all warehouses are sequenced according to the order of the holding cost only, but the ordered can be further modified by considering the time and cost of transportation also. Some possible future research is to study the various types of demand such as various types of demand like uncertain demand, probabilistic demand, time-dependent or price-dependent demand, etc. The model can be further extended to consider a supply chain system with multiple buyers and defective items. For further research, the model can be to incorporate the assumption, such as quantity discount, two-level trade credit policy, variable inflation, imperfect quality product, variable deterioration, etc. Moreover, the present idea can be extended to include other imprecise environments like fuzzy, random, fuzzy-random, bifuzzy, type-2 fuzzy, etc.