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.
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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.
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.
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\(k=\) number of warehouses.
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\(W_{ji}=\) the fixed storage capacity of jth (\(j=1, 2, 3, \ldots , k\)) warehouse for ith item.
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\(H_{j}=\) unit holding cost per unit item per unit time for jth (\(j=1, 2, 3, \ldots , k\)) warehouse.
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\(D_i(t)=\) demand rate at time t for ith item.
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\(\delta _i(t)=\) 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.
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\(\theta _i=\) deterioration rate of the ith item.
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\(r=\) the discount rate.
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\(f=\) the inflation rate, which is varied by the social economical situations.
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\(R=(r-f)=\) representing the net discount rate of inflation.
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\(B_{i}(t)=\) backlogged level at any time t(units) for ith item.
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\(c^{l}_{i}=\) unit opportunity cost due to lost sale for ith item, if the shortage is lost.
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\(\text {SC}_i=\) fixed cost of placing an order for ith item.
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\(t^{d}_i=\) time period during which no deterioration occurs for ith item (time units).
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\(c^{b}_{i}=\) unit backlogging cost per unit item per unit time for ith item, if the shortage is backlogged.
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\(s_{i}=\) unit selling price for ith item.
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\(p_i=\) unit purchasing cost for ith item.
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\(\text {TC}_i=\) total transportation cost for ith item.
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\(\text {TT}^{k-j+1}=\) total units of ith item transferred from \((k-j+ 1)\)th (\(j = 1, 2, 3, \ldots , k-1\)) to 1st warehouse.
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\(\text {TCP}^{k-j+1}_{i}=\) 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.
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\(d_{k-j+1}=\) distance of \((k-j+1)\)th warehouse from 1st warehouse.
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\(\text {TC}^{1}_{i}=\) fixed transportation cost for ith item.
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\(\text {TP}_i=\) the profit per unit time for ith item
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\(A_i=\) required storage area per unit quantity for ith item.
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\(A=\) total available storage space.
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\(B=\) available total budget cost.
Decision variables
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\(T_i=\) the length of the replenishment cycle of ith item.
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\(T_{ji}=\) 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
The upper \((\overline{\zeta })\) and lower \((\underline{\zeta })\) approximations of the rough variable \(\zeta \) are defined as follows:
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
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:
Example 3.2
Let us consider the triangular fuzzy variable \(\zeta \) with the following membership function
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
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
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
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
and
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
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
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.
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:
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
Now from Eqs. (10) and (11), we get
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
Afterward, the inventory depletes only due to deterioration up to time \(t=T_{ji}\); the differential equation is
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
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:
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
Now from Eqs. (17) and (18), we get
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
The present value of the deterioration cost for ith item is
The present value of the backlogging cost and the opportunity cost due to lost sale for ith item are
and
The present worth of the revenue for ith item is
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
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
The present value of the total transportation cost for ith item is
The present worth of the purchasing cost for ith item is
Therefore, the retailer’s average total profit (\(\text {TP}\)) is given by
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.
where
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:
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:
and
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:
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:
-
(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.
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(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.
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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.
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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.
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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.
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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.
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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.
References
Alturki I, Alfares H (2019) Optimum inventory control and warehouse selection with a time-dependent selling price. Ind Syst Eng Conf 1:1–6
Banerjee S, Agrawal S (2008) A two-warehouse inventory model for items with three-parameter Weibull distribution deterioration, shortages and linear trend in demand. Int Trans Oper Res 15:755–775
Bhunia AK, Maiti M (1998) A two warehouse inventory model for deteriorating items with a linear trend in demand and shortages. J Oper Res Soc 49:287–292
Bhunia AK, Shaikh AA (2016) Investigation of two-warehouse inventory problems in interval environment under inflation via particle swarm optimization. Math Comput Model Dyn Syst 22:160–179
Bishi B, Behera J, Sahu SK (2019) Two-warehouse inventory model for non-instantaneous deteriorating items with exponential demand rate. Int J Appl Eng Res 14:495–515
Brahmbhatt AC (1982) Economic order quantity under variable rate of inflation and mark-up prices. Productivity 23:127–130
Buzacott JA (1975) Economic order quantities with inflation. Oper Res Q 26:553–558
Castillo O (2018) Towards finding the optimal n in designing type- n fuzzy systems for particular classes of problems: a review. Appl Comput Math 17:3–9
Chakrabarty R, Roy T, Chaudhuri KS (2017) A production: inventory model for defective items with shortages incorporating inflation and time value of money. Int J Appl Comput Math 3:195–212
Chakrabarty R, Roy T, Chaudhuri KS (2018) A two-warehouse inventory model for deteriorating items with capacity constraints and back-ordering under financial considerations. Int J Appl Comput Math 4:1–16
Chakraborty D, Jana DK, Roy TK (2014) A new approach to solve intuitionistic fuzzy optimization problem using possibility, necessity, and credibility measures. Int J Eng Math 1:1–12
Chakraborty D, Jana DK, Roy TK (2015a) A new approach to solve multi-objective multi-choice multi-item Atanassov’s intuitionistic fuzzy transportation problem using chance operator. J Intell Fuzzy Syst 28:843–865
Chakraborty D, Jana DK, Roy TK (2015b) Multi-item integrated supply chain model for deteriorating items with stock dependent demand under fuzzy random and bifuzzy environments. Comput Ind Eng 88:166–180
Chakraborty D, Jana DK, Roy TK (2016) Expected value of intuitionistic fuzzy number and its application to solve multi-objective multi-item solid transportation problem for damageable items in intuitionistic fuzzy environment. J Intell Fuzzy Syst 30:1109–1122
Chakraborty D, Garai T, Jana DK, Roy TK (2017) A three-layer supply chain inventory model for non-instantaneous deteriorating item with inflation and delay in payments in random fuzzy environment. J Ind Prod Eng 34:407–424
Chakraborty D, Jana DK, Roy TK (2018) Two-warehouse partial backlogging inventory model with ramp type demand rate, three-parameter Weibull distribution deterioration under inflation and permissible delay in payments. Comput Ind Eng 123:157–179
Chang C-T, Teng J-T, Goyal SK (2010) Optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand. Int J Prod Econ 123:62–68
Chen J-M (1998) An inventory model for deteriorating items with time-proportional demand and shortages under inflation and time discounting. Int J Prod Econ 55:21–30
Choi J, Cao JJ, Romeijn EH, Geunes J, Bai SX (2005) A stochastic multi-item inventory model with unequal replenishment intervals and limited warehouse capacity. IIE Trans 37:1129–1141
Chung K-J, Her C-C, Lin S-D (2009) A two-warehouse inventory model with imperfect quality production processes. Comput Ind Eng 56:193–197
Covert RP, Philip GC (1973) An EOQ model for items with Weibull distribution deterioration. AIIE Trans 5:323–326
Das D, Roy A, Kar S (2010) Improving production policy for a deteriorating item under permissible delay in payments with stock-dependent demand rate. Comput Math Appl 60:1973–1985
Das D, Kar MB, Roy A, Kar S (2012) Two-warehouse production model for deteriorating inventory items with stock-dependent demand under inflation over a random planning horizon. CEJOR 20:251–280
Das D, Roy A, Kar S (2015) A multi-warehouse partial backlogging inventory model for deteriorating items under inflation when a delay in payment is permissible. Ann Oper Res 226:133–162
Dipana M, Kumar J, Shankar R, Goswami A (2016) A two-warehouse inventory model for non-instantaneous deteriorating items over stochastic planning horizon. J Ind Prod Eng 33:516–532
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Fuzzy Sets Syst 17:191–208
Ghare PM, Schrader GP (1963) A model for exponential decaying inventory. J Ind Eng 14:238–243
Goswami A, Chaudhuri KS (1992) An economic order quantity model for items with two levels of storage for a linear trend in demand. J Oper Res Soc 43:157–167
Guchhait P, Maiti MK, Maiti M (2014) Inventory policy of a deteriorating item with variable demand under trade credit period. Comput Ind Eng 76:75–88
Guria A, Das B, Mondal S, Maiti M (2013) Inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment. Appl Math Model 37:240–257
Hartely RV (1976) Operations research a managerial emphasis. Good Year Publishing Company, Santa Monica, CA, pp 315–317
Hwang H, Sohn KI (1983) Management of deteriorating inventory under inflation. Eng Econ 28:191–206
Indrajitsingha SK, Samanta PN, Misra UK (2019) A fuzzy two-warehouse inventory model for single deteriorating item with selling-price-dependent demand and shortage under partial-backlogged condition. Appl Appl Math 14:511–536
Jaggi CK, Verma P (2010) An optimal replenishment policy for non-instantaneous deteriorating items with two storage facilities. Int J Serv Oper Inf 5:209–230
Jaggi CK, Khanna A, Verma P (2011) Two-warehouse partial backlogging inventory model for deteriorating items with linear trend in demand under inflationary conditions. Int J Syst Sci 42:1185–1196
Jaggi CK, Verma P, Gupta M (2015) Ordering policy for non-instantaneous deteriorating items in two warehouse environment with shortages. Int J Logist Syst Manag 22:103–124
Jaggi CK, Cárdenas-Barrón LE, Tiwari S, Shafi AA (2017) Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments. Sci Iran E 24:390–412
Jana DK, Maity K, Roy TK (2013) Multi-objective imperfect production inventory model in fuzzy rough environment via genetic algorithm. Int J Oper Res 18:365–385
Jana DK, Maity K, Maiti M, Roy TK (2014) A multiobjective multi-item inventory control problem in fuzzy-rough environment using soft computing techniques. Adv Decis Sci 2014:1–13
Khan MAA, Shaikh AA, Panda GC, Konstantaras I (2019) Two-warehouse inventory model for deteriorating items with partial backlogging and advance payment scheme. RAIRO Oper Res 53:1691–1708
Kumar A, Chanda U (2018) Two-warehouse inventory model for deteriorating items with demand influenced by innovation criterion in growing technology market. J Manag Anal 1:1–15
Lee Y, Dye CY (2012) An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate. Comput Ind Eng 63:474–482
Lee CC, Hsu S-L (2009) A two-warehouse production model for deteriorating inventory items with time-dependent demands. Eur J Oper Res 194:700–710
Liang Y, Zhou F (2011) A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment. Appl Math Model 35:2221–2231
Maihami R, Karimi B (2014) Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts. Comput Oper Res 51:302–312
Mandal P, Giri BC (2019) A two-warehouse integrated inventory model with imperfect production process under stock-dependent demand and quantity discount offer. Int J Syst Sci Oper Logist 6:15–26
Manna AK, Dey JK, Mondal SK (2014) Three-layer supply chain in an imperfect production inventory model with two storage facilities under fuzzy rough environment. J Uncertain Anal Appl 2:1–31
Misra RB (1979) A note on optimal inventory management under inflation. Nav Res Logist Q 26:161–165
Mondal M, Maity AK, Maiti MM, Maiti M (2013) A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money. Appl Math Model 37:3200–3215
Morsi NN, Yakout MM (1998) Axiomatics for fuzzy rough sets. Fuzzy Sets Syst 100:327–342
Olivas F, Valdez F, Castillo O, Gonzalez CI, Martinez G, Melin P (2017) Ant colony optimization with dynamic parameter adaptation based on interval type-2 fuzzy logic systems. Appl Soft Comput 53:74–87
Ouyang LY, Wu KS, Yang CT (2006) A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Comput Ind Eng 51:637–651
Pakkala TPM, Achary KK (1992) A deterministic inventory model for deteriorating items with two warehouses and finite replenishment rate. Eur J Oper Res 57:71–76
Palanivel M, Uthayakumar R (2017) Two-warehouse inventory model for non-instantaneous deteriorating items with partial backlogging and permissible delay in payments under inflation. Int J Oper Res 28:35–69
Panda GC, Khan MAA, Shaikh AA (2019) A credit policy approach in a two-warehouse inventory model for deteriorating items with price-and stock-dependent demand under partial backlogging. J Ind Eng Int 15:147–170
Philip GC (1974) A generalized EOQ model for items with Weibull distribution. AIIE Trans 6:159–162
Radzikowska AM, Kerre EE (2002) A comparative study of rough sets. Fuzzy Sets Syst 126:137–155
Rong M, Mahapatra NK, Maiti M (2008) A two warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time. Eur J Oper Res 189:59–75
Roy A, Kar S, Maiti M (2008) A deteriorating multi-item inventory model with fuzzy costs and resources based on two different defuzzification techniques. Appl Math Model 32:208–223
Sarma KV (1983) A deterministic inventory model with two levels of storage and an optimum release rule. Opsearch 20:175–180
Shaikh AA, Cárdenas-Barrón LE, Tiwari S (2017) A two-warehouse inventory model for non-instantaneous deteriorating items with interval-valued inventory costs and stock-dependent demand under inflationary conditions. Neural Comput Appl 1:1–18
Shaikh AA, Cárdenas-Barrón LE, Tiwari S (2019) A two-warehouse inventory model for non-instantaneous deteriorating items with interval-valued inventory costs and stock-dependent demand under inflationary conditions. Neural Comput Appl 31:1931–1948
Soni HN (2013) Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment. Int J Prod Econ 146:259–268
Soni HN, Suthar DN (2018) Pricing and inventory decisions for non-instantaneous deteriorating items with price and promotional effort stochastic demand. J Control Decis 1:1–25
Taleizadeh AA, Wee H-M, Jolai F (2013) Revisiting a fuzzy rough economic order quantity model for deteriorating items considering quantity discount and prepayment. Math Comput Model 57:1466–1479
Tiwari S, Cárdenas-Barrón LE, Khanna A, Jagg CK (2016) Impact of trade credit and inflation on retailer’s ordering policies for non-instantaneous deteriorating items in a two-warehouse environment. Int J Prod Econ 176:154–169
Tiwari S, Jaggi CK, Bhunia AK, Shaikh AA, Goh M (2017) Two-warehouse inventory model for non-instantaneous deteriorating items with stock-dependent demand and inflation using particle swarm optimization. Ann Oper Res 254:401–423
Valdez F, Castillo O, Jain A, Jana DK (2019) Nature-inspired optimization algorithms for neuro-fuzzy models in real-world control and robotics applications. Comput Intell Neurosci. https://doi.org/10.1155/2019/9128451
Valliathal M, Uthayakumar R (2011) Optimal pricing and replenishment policies of an EOQ model for non-instantaneous deteriorating items with shortages. Int J Adv Manuf Technol 54:361–371
Wee H-M (1995) A deterministic lot-size inventory model for deteriorating items with shortages and a declining market. Comput Oper Res 22:345–356
Widyadana GA, Cárdenas-Barrón LE, Wee HM (2011) Economic order quantity model for deteriorating items and planned backorder level. Math Comput Model 54:1569–1575
Wu K-S (2003) The effect of inflation and time discounting on inventory replenishment model for deteriorating item with time-varying demand and shortages in all cycles. J Stat Manag Syst 6:519–536
Wu KS, Ouyang LY, Yang CT (2006) An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. Int J Prod Econ 101:369–384
Xu J, Zhao L (2008) A class of fuzzy rough expected value multi-objective decision making model and its application to inventory problems. Comput Math Appl 56:2107–2119
Xu J, Zhao L (2010) A multi-objective decision-making with fuzzy rough coefficients and its application to the inventory to the inventory problem. Inf Sci 180:679–696
Xu J, Zhou X (2009) Fuzzy-like multiple objective decision making. Springer, Berlin
Zhou Y-W (2003) A multi-warehouse inventory model for items with time-varying demand and shortages. Comput Oper Res 30:2115–2134
Zhou Y-W, Yang S-L (2005) A two-warehouse inventory model for items with stock-level-dependent demand rate. Int J Prod Econ 95:215–228
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Chakraborty, D., Jana, D.K. & Roy, T.K. Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment. Soft Comput 24, 14471–14490 (2020). https://doi.org/10.1007/s00500-020-04800-3
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DOI: https://doi.org/10.1007/s00500-020-04800-3