# A fuzzy rough multi-objective multi-item inventory model with both stock-dependent demand and holding cost rate

• Totan Garai
• Dipankar Chakraborty
• Tapan Kumar Roy
Original Paper

## Abstract

In this paper, we developed a multi-objective multi-item inventory model with fuzzy rough coefficients. Here, we have considered both demand and holding cost which is a non-linear function of the instantaneous stock level. Chance-constrained fuzzy rough multi-objective model and a traditional solution procedure based on an interactive fuzzy satisfying method are discussed. By examining the various definitions and theoretical results of fuzzy rough variables, we have designed a Tr-Pos chance constrained technique to determine the optimal solutions of a fuzzy rough multi-objective inventory problem. Finally, a numerical example is provided to illustrate the present model, and a sensitivity analysis of the optimal solution with respect to the major parameters is carried out.

## Keywords

Multi-objective Multi-item inventory Stock-dependent demand Stock-dependent holding cost Fuzzy rough variable Chance measure

## 1 Introduction

Different types of uncertainty such as fuzziness, roughness and randomness are common factors in many real life problems. Since Zadeh (1965) introduced the fuzzy set in 1965, fuzzy set theory has been well developed and employed to an extensive variety of real problems (Ishii and Konno 1998). Possibility theory was also proposed by Zadeh (1978) and developed by many researchers such as Dubois and Prade (1988). However, in a decision-making process, we may face a hybrid uncertain environment where fuzziness and roughness be present at the same time. In such cases, a fuzzy rough variable is a useful tool. Fuzziness and roughness play a significant role among types of uncertain problems. The concept of fuzzy rough sets first introduced by Dubois and Prade (1990) plays a key role dealing with the two types of uncertainty simultaneously. Nowadays, many researchers have considered the issue of combing fuzziness and roughness in a general framework for the study of fuzzy rough sets. Some definitions and valuable properties of the fuzzy rough variable presented by Liu (2002). At present, using these approaches some researches (Mondal et al. 2013a; Xu and Zhao 2010; Maiti and Maiti 2005; Pedrycz and Chen 2011, 2015a; Alfares and Ghaithan 2013; Khouja 1995; Garai et al. 2017a; Maity 2011; Tsai et al. 2012) modelled different practical problems where both fuzziness and roughness exist simultaneously.

In numerous cases, it is established that the parameters of some inventory problems are considered fuzzy and rough uncertainties. For example, production cost, set-up cost, holding cost, repairing cost, etc. realise on various factors such as inflation, labour travail wages, wear and tear cost, bank interest, etc. which are uncertain in fuzzy rough sense. To be more specific, set-up cost depends on the total quantity to be produced in a scheduling period, and the inventory holding cost of an item is supposed to be dependent on the amount of storage. Moreover, with the inventory, the total quantity to be produced in a scheduling period and the amount storage may be uncertain. This uncertainty may consider in fuzzy environment. In these circumstances, fuzzy rough can be used for the formulation of inventory problems. In the literature, very few researchers (Xu and Zaho 2008; Xu and Zhao 2010; Chen and Chung 2006; Horng et al. 2005; Chen and Kao 2013; Chen 1996; Lushu and Nair 2002; Li 2005) developed and solved inventory or production–inventory problems with the fuzzy-rough environment.

For the inventory problem, the classical inventory decision-making models have deliberated a single item. However, single item inventories rarely occur, whereas multi-item inventories are common in real-life circumstances. Many researchers (cf. Balkhi and Foul 2009; Hartley 1978; Lee and Yao 1998; Taleizadeh et al. 2011) investigated the multi-item inventory models under resource constraints. The inventory problem is an issue that has received considerable attention in inventory models with different types of demand rates. Gupta and Vrat (1986) were the first researcher to develop a multi-item inventory model with stock-dependent consumption rates. A replenishment model for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging developed by Wu et al. (2006). Other studies in this area include those of Avinadav et al. (2013), Chen and Chien (2011a, b), Morsi and Yakout (1998), Mondal et al. (2013b), Garai et al. (2017b), Min et al. (2012) and Taleizadeh et al. (2013).

In real life, the holding cost of perishable booze such as vegetables, fruit, milk and foodstuffs drop with each going day, and increasing holding costs are essential to maintaining the freshness of the goods and to obstruct spoilage. Many inventory models assume the unit holding cost to be variable. Pando et al. (2012) developed an inventory model with both the demand rate and holding cost dependent on the stock level. Recently, examples of other studies in this area include Tripathi (2013), Chen et al. (2001), Chen and Chang (2001), Radzikowska and Kerre (2002), Pedrycz and Chen (2015b) and Roy (2008). The summary of some related literature for inventory models are represented in Table 1.
Table 1

Summary of related literature for inventory model

Methods

Item

Objective

Demand rate

Holding cost

Shortage

Environment

Pando et al. (2012)

Single

Single

Stock-dependent (power function)

Nonlinear-stock dependent

No

Crisp

Single

Single

Constant

Constant

Completely backlogged

Crisp

Tripathi (2013)

Single

Single

Time-dependent (power function)

Linear-time dependent

No

Crisp

Xu and Zaho (2008)

Multi

Multi

Constant

Constant

No

Fuzzy rough

Mondal et al. (2013b)

Multi

Single

Stock-dependent (linear function)

Constant

No

Fuzzy rough

Xu and Zhao (2010)

Multi

Multi

Constant

Constant

No

Fuzzy rough

Pando et al. (2013)

Single

Single

Stock-dependent (power function)

Nonlinear stock and time dependent

No

Crisp

Min et al. (2012)

Single

Single

Stock-dependent (power function)

Constant

No

Crisp

Yang (2014)

Single

Single

Stock-dependent (power function)

Nonlinear-stock dependent

Partially backlogged

Crisp

Pando et al. (2012)

Multi

Single

Price-dependent (power function)

Linear-time dependent

No

Crisp

This proposed method

Multi

Multi

Stock-dependent (power function)

Nonlinear-stock dependent

Partially backlogged

Fuzzy rough

In spite of the above-mentioned developments, following additions can also be made in fuzzy rough numbers, the formulation and solution of a multi-objective multi-item inventory model under fuzzy rough environments.
• Tr-Pos constrained multi-objective model with fuzzy rough variables.

• Multi-objective multi-item inventory model with demand is a power function and holding cost is a non-linear function of the stock level.

• Multi-objective multi-item inventory model under fuzzy rough environments.

The rest of the paper is organized as follows: in Sect. 2, we present some basic knowledge of fuzzy rough theory and optimization theory. In Sect. 3, we study the Tr-Pos constrained multi-objective programming model with fuzzy-rough variable, and an interactive fuzzy satisfying method is adopted to obtain a satisfactory solution for the decision maker. Sect. 4, provides the notations and assumptions which are used throughout the paper. In Sect. 5, a multi-objective multi-item inventory model has been developed in the fuzzy-rough environment and discuss its solution method. Numerical example to illustrate the models are provided in Sect. 6. In Sect. 7, the result of the change of different parameters is discussed graphically. Finally, the conclusion and scope of the future work plan have been made in Sect. 8.

## 2 Preliminaries and deductions

In this section, we recall some concepts and properties of fuzzy variables, rough variables, and fuzzy rough variables, which will be applied in the following sections.

## Definition 2.1

(Xu and Zaho 2008) Let $$\Theta$$ be a non-empty set, $$\mathscr {P}(\Theta )$$ be the power set of $$\Theta$$, and Pos a possibility measure. The triplet $$(\Theta , \mathscr {P}(\Theta ), Pos )$$ is called a possibility space. A fuzzy variable is defined as a function from a possibility space $$(\Theta , \mathscr {P}(\Theta ), Pos )$$ to the real line $$\mathbb {R}$$.

The credibility theory is a branch of mathematics that studies the behaviour of fuzzy phenomena.

## Definition 2.2

(Liu 2002) Let $$(\Theta , \mathscr {P}(\Theta ), Pos )$$ be a possibility space, and A a set in $$\mathscr {P}(\Theta )$$. Then the credibility measure of A is defined by
\begin{aligned} Cr\{A\}=\dfrac{1}{2}\left( Pos\{A\} + Nec\{A\}\right) \end{aligned}
where Pos and Nec represent the possibility measure and the necessity measure, respectively.

Trust theory is a branch of mathematics that studies the behaviour of rough event.

## Definition 2.3

(Liu 2002) Let $$\Lambda$$ be a non-empty set (Tsai et al. 2008), $$\mathscr {A}$$ a $$\sigma$$-algebra of subsets of $$\Lambda$$, $$\Delta$$ an element in $$\mathscr {A}$$, and $$\pi$$ a set function satisfying the following conditions:
1. (i)

$$\pi \{\Lambda \} < +\infty$$;

2. (ii)

$$\pi \{\Delta \}> 0$$;

3. (iii)

$$\pi \{A\}>0$$ for any $$A \in \mathscr {A}$$;

4. (iv)

For every countable sequence of mutually disjoint events $$\{A_{i}\}_{i}^{\infty }$$, we have $$\pi \left\{ \bigcup _{i=1}^{\infty }\right\} A_{i} = \sum _{i=1}^{\infty } \pi \{A_{i}\}$$

Then $$(\Lambda , \Delta , \mathscr {A}, \pi )$$ is called a rough space.

## Definition 2.4

(Xu and Zaho 2008) 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 $$B$$ of $$\mathbb {R}$$, we have
\begin{aligned} \{\eta \in \Lambda : \zeta (\eta )\in 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 \}\quad \underline{\zeta }=\{\zeta (\eta ) : \eta \in \Delta \} \end{aligned}

## Definition 2.5

(Xu and Zaho 2008) Let $$(\Lambda , \Delta , \mathscr {A}, \pi )$$ be a rough space. The trust measure of the event A is defined by
\begin{aligned} Tr\{A\}=\dfrac{1}{2}(\underline{Tr}\{A\} + \overline{Tr}\{A\}) \end{aligned}
where the upper trust measure $$\overline{Tr}\{A\} = \dfrac{\pi \{A\}}{\pi \{\Lambda \}}$$ and lower trust measure $$\underline{Tr}\{A\} = \dfrac{\pi \{A \cap \Delta \}}{\pi \{\Delta \}}$$.

When we do not have enough information to determine the measure $$\pi$$ for a real-life problem, we can consider that all elements in $$\Lambda$$ are equally likely to occur. For this case, the measure $$\pi$$ may be treated as the Lebesgue measure.

## Example 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 the Definitions 2.4 and 2.5, we can obtain the trust measure of the event $$\{\zeta \ge t\}$$ and and $$\{\zeta \le t\}$$ as follows (Fig. 1):
\begin{aligned} Tr(\zeta \ge t)=\left\{ \begin{array}{ll} 0 &\quad {\text {if}}\,\, b_{2} \le t \\ \dfrac{b_{2}-t}{2(b_{2}-b_{1})} &\quad {\text {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 {\text {if}}\,\,a_{1} \le t \le a_{2} \\ \dfrac{1}{2}\left( \dfrac{b_{2}-t}{b_{2}-b_{1}} + 1\right) &\quad {\text {if}}\,\, b_{1} \le t \le a_{1} \\ 1 &\quad {\text {if}}\,\, t \le b_{1} \end{array}\right. \end{aligned}
(1)
and
\begin{aligned} Tr(\zeta \le t)=\left\{ \begin{array}{ll} 0 &\quad {\text {if}}\,\, t \le b_{1} \\ \dfrac{t-b_{1}}{2(b_{2}-b_{1})} &\quad {\text {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 {\text {if}}\,\, a_{1} \le t \le a_{2} \\ \dfrac{1}{2}\left( \dfrac{t -b_{1}}{b_{2}-b_{1}} + 1\right) &\quad {\text {if}}\,\, a_{2} \le t \le b_{2} \\ 1 &\quad {\text {if}}\,\, b_{2} \le t \end{array}\right. \end{aligned}
(2)

## Definition 2.6

(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 $$Pos\{\zeta (\eta ) \in B \}$$ is a measurable function of $$\eta$$ for any Borel set $$B$$ of $$\mathbb {R}$$. Usually, say that a fuzzy rough variable is a rough variable taking fuzzy values.

## Example 2

Let’s consider the triangular fuzzy variable $$\tilde{a}=(\zeta _{1}, \zeta _{2}, \zeta _{3})$$ with the following membership function
\begin{aligned} \mu _{\tilde{a}}(t) = \left\{ \begin{array}{ll} \dfrac{t-\zeta _{1}}{\zeta _{2}-\zeta _{1}} &\quad \,\,{\text {if}}\,\, \zeta _{1}\le t<\zeta _{2} \\ 1 &\quad \,\,{\text {if}}\,\, t = \zeta _{2} \\ \dfrac{\zeta _{3}-t}{\zeta _{3}-\zeta _{2}} &\quad \,\,{\text {if}}\,\, \zeta _{2}< t \le \zeta _{3} \\ 0 &\quad \,{\text {otherwise}} \end{array}\right. \end{aligned}
(3)
where, every $$\zeta _{i}$$ is a positive real number for $$i=1,2,3$$. Now, we assume that every $$\zeta _{i} \vdash ([a_{1}, a_{2}] [b_{1}, b_{2}])$$ is rough variable for $$i=1, 2, 3$$. Then $$\tilde{a}$$ is called a triangular fuzzy rough variable.

## Definition 2.7

(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 $$Pos\{\zeta (\eta ) \in B \}$$ is a measurable function of $$\zeta$$ for any Borel set $$B$$ of $$\mathbb {R}^{n}$$.

## Definition 2.8

(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 2.9

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

## Proposition 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 $$Pos\{g_{j}(\zeta (\eta )) \le 0, j=1, 2, \ldots , q\}$$ is a rough variable.

## 3 Tr-Pos constrained multi-objective model with fuzzy rough variable

Fuzzy programming of the multi-objective problem has been well manifested. It has been increasingly acknowledged that many real-world decision-making problems involve multiple and competing objectives which should be deliberated together. As a propagation of the fuzzy multi-objective decision-making case, the fuzzy rough multi-objective linear decision-making model (Xu and Zhao 2010) is defined as a means for optimizing multiple several objective functions subject to a number of constraints.

Consider the following fuzzy-rough multi-objective problem:
\begin{aligned} \begin{array}{ll} {\text {Max}} & \quad \left\{ \widehat{\tilde{c}}_{1}^{\text {T}}x, \widehat{\tilde{c}}_{2}^{\text {T}}x, \ldots , \widehat{\tilde{c}}_{p}^{\text {T}}x \right\} \\ \text {s.t.} & \quad \widehat{\tilde{a}}_{j}^{\text {T}} x \le \widehat{\tilde{b}}_j; \,\, j=1, 2, \ldots , q, \\ & \quad x \ge 0, \\ \end{array} \end{aligned}
(4)
where $$\widehat{\tilde{c}}_{i}=(\widehat{c}_{i1}, \widehat{c}_{i2}, \ldots , \widehat{c}_{in})^{\text {T}}$$, $$\widehat{\tilde{a}}_{j}=(\widehat{a}_{j1}, \widehat{a}_{j2}, \ldots , \widehat{a}_{jn})^{\text {T}}$$ are fuzzy rough vectors, and $$\widehat{\tilde{b}}_{j}$$ are fuzzy rough variables, $$i=1, 2, \ldots , p \, \& \,\quad j=1, 2, \ldots , q.$$
From the Definition 2.9, we know that
\begin{aligned}&Ch\left\{ \widehat{\tilde{a}}_{j}^{\text {T}}x \le \widehat{\tilde{b}}_{j}\right\} (\delta _{j}) \ge \sigma _{j} \Leftrightarrow Tr \left\{ \eta | Pos \{\widehat{\tilde{a}}_{j}(\eta )^{\text {T}}x \le \widehat{\tilde{b}}_{j}\} \ge \sigma _{j}\right\} \ge \delta _{j}, j=1,2, \ldots , q \end{aligned}
For given confidence levels $$\delta _{j}, \sigma _{j}$$ using the primitive chance measure we have the constaints as follows:
\begin{aligned} Tr \{\eta | Pos \{\widehat{\tilde{a}}_{j}(\eta )^{\text {T}}x \le \widehat{\tilde{b}}_{j}\} \ge \sigma _{j}\} \ge \delta _{j}, \,\,\, j=1,2, \ldots , q \end{aligned}
Thus, a point $$x(\ge 0)$$ is called feasible for Eq. (4) if and only if the trust measures of the rough events $$\{\eta | Pos \{\widehat{\tilde{a}}_{j}(\eta )^{\text {T}}x \le \widehat{\tilde{b}}_{j}\} \ge \sigma _{j}\}$$ are at least $$\delta _{j}, \,\, j=1,2,\ldots ,q$$. Because
\begin{aligned} Ch\left\{ \widehat{\tilde{c}}_{i}(\eta )^{\text {T}}x \ge f_{i}\right\} (\alpha _{i}) \ge \beta _{i} \Leftrightarrow Tr \left\{ \eta | Pos\{\widehat{\tilde{c}}_{i}(\eta )^{\text {T}}x \ge f_{i}\} \ge \beta _{i}\right\} \ge \alpha _{i}, \,\,\, i=1,2,\ldots ,p \end{aligned}
then the chance-constrained multi-objective programming(CCMOP)model for Eq. (4) can be rewritten as:
\begin{aligned} \begin{array}{ll} {\text {Max}} & \quad \{f_{1}, f_{2}, \ldots , f_{p} \} \\ \text {s.t.} & \quad Tr \{\eta | Pos\{\widehat{\tilde{c}}_{i}(\eta )^{\text {T}}x \ge f_{i}\} \ge \beta _{i}\} \ge \alpha _{i}, \,\,\, i=1,2,\ldots ,p, \\ & \quad Tr \{\eta | Pos \{\widehat{\tilde{a}}_{j}(\eta )^{\text {T}}x \le \widehat{\tilde{b}}_{j}\} \ge \sigma _{j}\} \ge \delta _{j}, \,\, j=1,2, \ldots , q, \\ & \quad x \ge 0, \end{array} \end{aligned}
(5)
where $$\alpha _{i}, \beta _{i}, \sigma _{j}, \delta _{j}$$ are predetermined confidence levels for $$i=1,2,\ldots ,p; \, j=1,2,\ldots ,q$$, and abbreviations $$Tr\{*\}$$, $$Pos\{*\}$$ denotes the trust measure and possibility of the event in $$\{*\}$$ respectively.

## Proposition 2

Let $$\tilde{a}$$ and $$\tilde{b}$$ be two independent fuzzy numbers (Dubois and Prade 1988) with continuous membership function. For a given confident level $$\alpha \in [0, 1]$$
\begin{aligned} Pos\{\tilde{a} \ge \tilde{b}\} \ge \alpha \Leftrightarrow a_{\alpha }^{R} \ge b_{\alpha }^{L} \end{aligned}
\begin{aligned} Pos\{\tilde{a} \le \tilde{b}\} \ge \alpha \Leftrightarrow a_{\alpha }^{L} \le b_{\alpha }^{R} \end{aligned}
where $$a_{\alpha }^{L}, a_{\alpha }^{R}$$ and $$b_{\alpha }^{L}, b_{\alpha }^{R}$$ are the left and right side extreme points of the $$\alpha$$-level sets $$[a_{\alpha }^{L}, a_{\alpha }^{R}]$$ and $$[b_{\alpha }^{L}, b_{\alpha }^{R}]$$ of $$\tilde{a}$$ and $$\tilde{b}$$, respectively.

## Theorem 3.1

Let $$\widehat{\tilde{c}}_{ik}=(\widehat{c}_{ik}^{1}, \widehat{c}_{ik}^{2}, \widehat{c}_{ik}^{3})$$ be a triangular fuzzy rough variable, for any $$\eta \in \Lambda$$, the fuzzy variable $$\widehat{\tilde{c}}_{ik}(\eta )$$ is characterized by the following membership function
\begin{aligned} \mu _{\widehat{\tilde{c}}_{ik}(\eta )}(x) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{c}_{ik}^{1}}{\widehat{c}_{ik}^{2}-\widehat{c}_{ik}^{1}} &\quad \,\,{\text {if}}\,\, \widehat{c}_{ik}^{1}\le t \le \widehat{c}_{ik}^{2} \\ 1 &\quad \,\,{\text {if}}\,\, t=\widehat{c}_{ik}^{2} \\ \dfrac{\widehat{c}_{ik}^{3}-t}{\widehat{c}_{ik}^{3}-\widehat{c}_{ik}^{2}} &\quad \,\,{\text {if}}\,\, \widehat{c}_{ik}^{2} \le t \le \widehat{c}_{ik}^{3} \\ 0 &\quad \,{\text {otherwise}} \end{array}\right. \end{aligned}
where $$(c_{ik}(\eta ))_{n\times 1}=(c_{i1}(\eta ), c_{i2}(\eta ),\ldots , c_{in}(\eta ))^{\text {T}}$$ is a rough vector and $$\widehat{c}_{i}^{l}(\eta )^{\text {T}}x=([a_{1}, a_{2}] [b_{1}, b_{2}])$$(where $$b_{1} \le a_{1} < a_{2} \le b_{2}$$ and l=1,2,3) is a rough variable characterized by the following trust measure function
\begin{aligned} Tr(\widehat{c}_{i}(\eta )^{\text {T}}x \ge t)=\left\{ \begin{array}{ll}0 &\quad {\text {if}}\,\, b_{2} \le t \\ \dfrac{b_{2}-t}{2(b_{2}-b_{1})} &\quad {\text {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 {\text {if}}\,\, a_{1} \le t \le a_{2} \\ \dfrac{1}{2}\left( \dfrac{b_{2}-t}{b_{2}-b_{1}} + 1\right) &\quad {\text {if}}\,\, b_{1} \le t \le a_{1} \\ 1 &\quad {\text {if}}\,\, t \le b_{1} \end{array}\right. \end{aligned}
Then we have $$Tr\{\eta | Pos\{\widehat{\tilde{c}}_{i}(\eta )^{\text {T}} \ge f_{i}\}\ge \beta _{i}\} \ge \alpha _{i}$$, if and only if
\begin{aligned} \left\{ \begin{array}{ll} f_{i} \le b_{2}-2\alpha _{i}(b_{2}-b_{1})+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, a_{2} \le f_{i} -\beta _{i}c_{ij}^{r}x \le b_{2} \\ f_{i} \le \dfrac{b_{2}(a_{2}-a_{1})+a_{2}(b_{2}-b_{1})-2\alpha _{i}(b_{2}-b_{1})(a_{2}-a_{1})}{(b_{2}-b_{1})+(a_{2}-a_{1})}+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, a_{1} \le f_{i} -\beta _{i}c_{ij}^{r}x \le a_{2} \\ f_{i} \le b_{2}-(b_{2}-b_{1})(2\alpha _{i}-1)+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, b_{1} \le f_{i}-\beta _{i}c_{ij}^{r}x \le a_{1} \\ f_{i} \le b_{1}+ \beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, f_{i} - \beta _{i}c_{ij}^{r}x \le b_{1} \end{array}\right. \end{aligned}
(6)
where $$\alpha _{i}, \beta _{i} \in [0,1]$$ are predetermined confidence levels and $$c_{ij}^{r}$$ is the right spread of $$\widehat{\tilde{c}}_{ij}$$.

## Proof

From the assumption we know that $$(c_{ik}(\eta ))_{n\times 1} =(c_{i1}(\eta ), c_{i2}(\eta ),\ldots , c_{in}(\eta ))^{\text {T}}$$ and $$c_{ik}(\eta )$$ is a rough variable. Let $$c_{ik}(\eta )=([a_{1ik}, a_{2ik}], [b_{1ik}, b_{2ik}])$$ and $$x=(x_{1}, x_{2}, \ldots , x_{n})^{\text {T}}$$ then $$x_{k} c_{ik}(\eta )=([x_{k}a_{1ik}, x_{k}a_{2ik}], [x_{k}b_{1ik}, x_{k}b_{2ik}])$$
\begin{aligned} c_{i}(\eta )^{\text {T}}x&=\sum _{k=1}^{n}c_{ik}(\eta )x_{k}=\sum _{k=1}^{n}([x_{k}a_{1ik}, x_{k}a_{2ik}], [x_{k}b_{1ik}, x_{k}b_{2ik}])\\&=\left( \left[ \sum _{k=1}^{n}x_{k}a_{1ik}, \sum _{k=1}^{n}x_{k}a_{2ik}\right] , \left[ \sum _{k=1}^{n} x_{k}b_{1ik}, \sum _{k=1}^{n}x_{k}b_{2ik}\right] \right) \end{aligned}
Therefore, $$c_{i}(\eta )^{\text {T}}x$$ is also a rough variable, Now we can assume that $$a_{1}=\sum _{k=1}^{n}x_{k}a_{1ik}$$, $$a_{2}=\sum _{k=1}^{n}x_{k}a_{2ik}$$ $$b_{1}=\sum _{k=1}^{n}x_{k}b_{1ik}$$, $$b_{2}=\sum _{k=1}^{n}x_{k}b_{2ik}$$ Then $$c_{i}(\eta )^{\text {T}}x=([a_{1}, a_{2}], [b_{1}, b_{2}])$$ Moreover, we know that $$\widehat{\tilde{c}}_{ik}(\eta )$$ is a fuzzy number with the membership function $$\mu _{\widehat{\tilde{c}}_{ik}(\eta )}$$ for any $$\eta \in \Lambda$$. The fuzzy number $$\widehat{\tilde{c}}_{ik}(\eta )^{\text {T}}$$ is characterized by the membership function in the following form
\begin{aligned} \mu _{\widehat{\tilde{c}}_{ik}(\eta )}(x) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{c}_{ik}^{1}}{\widehat{c}_{ik}^{2}-\widehat{c}_{ik}^{1}} &\quad \,\,{\text {if}}\,\, \widehat{c}_{ik}^{1}\le t \le \widehat{c}_{ik}^{2} \\ 1 &\quad \,\,{\text {if}}\,\, t=\widehat{c}_{ik}^{2} \\ \dfrac{\widehat{c}_{ik}^{3}-t}{\widehat{c}_{ik}^{3}-\widehat{c}_{ik}^{2}} &\quad \,\,{\text {if}}\,\, \widehat{c}_{ik}^{2} \le t \le \widehat{c}_{ik}^{3} \\ 0 &\quad \,{\text {otherwise}} \end{array}\right. \end{aligned}
where $$i=1,2, \ldots , p$$. By the Proposition 2, we have that $$Pos\{\widehat{\tilde{c}}_{i}(\eta )^{\text {T}} \ge f_{i}\}\ge \beta _{i} \Longleftrightarrow c_{i}(\eta )^{\text {T}}x + \beta _{i}c_{ij}^{r}x \ge f_{i}$$, for $$i=1,2,\ldots ,p$$
For the given confidence level $$\beta _{i}, \alpha _{i} \in [0, 1]$$, we have $$Tr\{\eta | Pos\{\widehat{\tilde{c}}_{i}(\eta )^{\text {T}} \ge f_{i}\}\ge \beta _{i}\} \ge \alpha _{i} \Longleftrightarrow Tr\{\eta | c_{i}(\eta )^{\text {T}}x \ge f_{i} - \beta _{i}c_{ij}^{r}x\} \ge \alpha _{i}$$
\begin{aligned} \Longleftrightarrow \left\{ \begin{array}{ll} f_{i} \le b_{2}-2\alpha _{i}(b_{2}-b_{1})+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, a_{2} \le f_{i} -\beta _{i}c_{ij}^{r}x \le b_{2} \\ f_{i} \le \dfrac{b_{2}(a_{2}-a_{1})+a_{2}(b_{2}-b_{1})-2\alpha _{i}(b_{2}-b_{1})(a_{2}-a_{1})}{(b_{2}-b_{1})+(a_{2}-a_{1})}+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, a_{1} \le f_{i} -\beta _{i}c_{ij}^{r}x \le a_{2} \\ f_{i} \le b_{2}-(b_{2}-b_{1})(2\alpha _{i}-1)+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, b_{1} \le f_{i}-\beta _{i}c_{ij}^{r}x \le a_{1} \\ f_{i} \le b_{1}+ \beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, f_{i} - \beta _{i}c_{ij}^{r}x \le b_{1} \end{array}\right. \end{aligned}
This completes the proof. $$\square$$

## Theorem 3.2

Assume that $$\widehat{\tilde{a}}_{jk}, \widehat{\tilde{b}}_{k}$$ are triangular fuzzy rough variables, for any $$\eta \in \Lambda$$, fuzzy variables $$\widehat{\tilde{a}}_{jk}, \widehat{\tilde{b}}_{k}$$ are characterized by the following membership functions
\begin{aligned} \mu _{\widehat{\tilde{a}}_{jk}(\eta )}(t) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{a}_{jk}^{1}}{\widehat{a}_{jk}^{2}-\widehat{a}_{jk}^{1}} &\quad \,\,{\text {if}}\,\, \widehat{a}_{jk}^{1}\le t \le \widehat{a}_{jk}^{2} \\ 1 &\quad \,\,{\text {if}}\,\, t=\widehat{a}_{jk}^{2} \\ \dfrac{\widehat{a}_{jk}^{3}-t}{\widehat{a}_{jk}^{3}-\widehat{a}_{jk}^{2}} &\quad \,\,{\text {if}}\,\, \widehat{a}_{jk}^{2} \le t \le \widehat{a}_{jk}^{3} \\ 0 &\quad \,{\text {otherwise}} \end{array}\right. \end{aligned}
and
\begin{aligned} \mu _{\widehat{\tilde{b}}_{j}(\eta )}(t) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{b}_{j}^{1}}{\widehat{b}_{j}^{2}-\widehat{b}_{j}^{1}} &\quad \,\,{\text {if}}\,\, \widehat{b}_{j}^{1}\le t \le \widehat{b}_{j}^{2} \\ 1 &\quad \,\,{\text {if}}\,\, t=\widehat{b}_{j}^{2} \\ \dfrac{\widehat{b}_{j}^{3}-t}{\widehat{b}_{j}^{3}-\widehat{b}_{j}^{2}} &\quad \,\,{\text {if}}\,\, \widehat{b}_{j}^{2} \le t \le \widehat{b}_{j}^{3} \\ 0 &\quad \,{\text {otherwise}} \end{array}\right. \end{aligned}
where $$(a_{jk}(\eta ))_{n\times 1}=(a_{k1}(\eta ), a_{k2}(\eta ),\ldots , a_{km}(\eta ))^{\text {T}}$$ is a rough vector, $$\widehat{a}_{jk}(\eta ), \widehat{b}_{j}(\eta )$$ are rough variables, $$j=1,2,\ldots ,p \,\,\, k=1,2,\ldots , m$$. By Theorem 3.1, we have $$\widehat{a}_{jk}(\eta )x, \widehat{b}_{j}(\eta )$$ are rough variables, then $$\widehat{a}_{j}(\eta )x - \widehat{b}_{j}(\eta )=([a_{1}, a_{2}] [b_{1}, b_{2}])$$ (where $$b_{1} \le a_{1} < a_{2} \le b_{2}$$ and $$l=1,2,3$$) is also a rough variable. We assume that it is characterized by the following trust measure function
\begin{aligned} Tr(\widehat{a}_{j}(\eta )x - \widehat{b}_{j}(\eta ) \le t)=\left\{ \begin{array}{ll}0 &\quad {\text {if}}\,\, t \le b_{1} \\ \dfrac{t-b_{1}}{2(b_{2}-b_{1})} &\quad {\text {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 {\text {if}}\,\, a_{1} \le t \le a_{2} \\ \dfrac{1}{2}\left( \dfrac{t -b_{1}}{b_{2}-b_{1}} + 1\right) &\quad {\text {if}}\,\, a_{2} \le t \le b_{2} \\ 1 &\quad {\text {if}}\,\, b_{2} \le t \end{array}\right. \end{aligned}
Then, we have that $$Tr\{\eta | Pos\{\widehat{\tilde{a}}_{j}(\eta )x \le \widehat{\tilde{b}}_{j}(\eta )\} \ge \sigma _{j}\} \ge \delta _{j}$$ if and only if
\begin{aligned} \left\{ \begin{array}{ll} L \ge b_{1}+2(b_{2}-b_{1})\delta _{j} &\quad {\text {if}}\,\, b_{1} \le L \le a_{1} \\ L \ge \dfrac{2 \delta _{j}(b_{2}-b_{1})(a_{2}-a_{1}) + b_{1}(a_{2}-a_{1})+a_{1}(b_{2}-b_{1})}{(a_{2}-a_{1})+(b_{2}-b_{1})} &\quad {\text {if}}\,\, a_{1} \le L \le a_{2} \\ L \ge (2\delta _{j}-1)(b_{2}-b_{1})+b_{1} &\quad {\text {if}}\,\, a_{2} \le L \le b_{2} \\ L \ge b_{2} &\quad {\text {if}}\,\, b_{2} \le L \end{array}\right. \end{aligned}
(7)
where $$L= \sigma _{j} a_{jk}^{l}x + \sigma _{j} b_{j}^{r}$$, and $$a_{jk}^{l}, b_{j}^{r}$$ are the left spread and right spread of $$\widehat{\tilde{a}}_{jk}, \widehat{\tilde{b}}_{j}$$ respectively.

## Proof

Proof is similar as Theorem 3.1. $$\square$$

From Theorems 3.1 and 3.2, we know that Eq. (5) is equivalent to the following multi-objective programming problem,
\begin{aligned} \text {Max }\,&\{f_{1}, f_{2}, \ldots , f_{p}\} \\ \text {s. t. }&\left\{ \begin{array}{ll} f_{i} \le b_{2}-2\alpha _{i}(b_{2}-b_{1})+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, a_{2} \le f_{i} -\beta _{i}c_{ij}^{r}x \le b_{2} \\ f_{i} \le \dfrac{b_{2}(a_{2}-a_{1})+a_{2}(b_{2}-b_{1})-2\alpha _{i}(b_{2}-b_{1})(a_{2}-a_{1})}{(b_{2}-b_{1})+(a_{2}-a_{1})}+\beta _{i}c_{ij}^{r} x &\quad {\text {if}}\,\, a_{1} \le f_{i} -\beta _{i}c_{ij}^{r}x \le a_{2} \\ f_{i} \le b_{2}-(b_{2}-b_{1})(2\alpha _{i}-1)+\beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, b_{1} \le f_{i}-\beta _{i}c_{ij}^{r}x \le a_{1} \\ f_{i} \le b_{1}+ \beta _{i}c_{ij}^{r}x &\quad {\text {if}}\,\, f_{i} - \beta _{i}c_{ij}^{r}x \le b_{1} \\ x \in X \end{array}\right. \end{aligned}
(8)

### 3.1 Fuzzy interactive satisfied method

Here, we introduce the interactive fuzzy satisfied method (FISM) proposed by Sakawa (1993). We consider the first case in the multi-objective programming Eq. (8) as our research objective and use the interactive fuzzy satisfying method to get an optimal solution to Eq. (5).

When $$a_{2} \le f_{i} -\beta _{i}c_{ij}^{r}x \le b_{2}$$, according to Theorems 3.1 and 3.2, Eq. (5) is equivalent to the following multi-objective programming problem
\begin{aligned} \text {Max }\,&\{f_{1}, f_{2}, \ldots , f_{p}\} \\ \text {s. t. }&\left\{ \begin{array}{ll} f_{i} \le b_{2}-2\alpha _{i}(b_{2}-b_{1})+\beta _{i}c_{ij}^{r}x \\ x \in X. \end{array}\right. \end{aligned}
(9)
or, equivalently
\begin{aligned} \text {Max }\,&\{G_{1}(x), G_{2}(x), \ldots , G_{p}(x)\} \\ \text {s. t. }&x \in X \end{aligned}
(10)
where $$G_{i}(x) = b_{2}-2\alpha _{i}(b_{2}-b_{1})+\beta _{i}c_{ij}^{r}x$$.
The objective function of Eq. (10) is to maximize $$G_{i}(x)$$, therefore, for each objective, we introduce the fuzzy goals $$G_{i}(x)$$ approximately larger than a certain value, and the fuzzy goal is to characterize by the following membership function
\begin{aligned} \mu _{i}(G_{i}(x)) \left\{ \begin{array}{ll} 1 &\quad {\text {if}}\,\, G_{i}(x)> G_{i}^{1} \\ 1- \dfrac{G_{i}(x) -G_{i}^{0}}{G_{i}^{1}-G_{i}^{0}} &\quad {\text {if}}\,\, G_{i}^{0} \le G_{i}(x) \le G_{i}^{1} \\ 0 &\quad {\text {if}}\,\, G_{i}(x) < G_{i}^{0} \end{array}\right. \end{aligned}
(11)
In Eq. (11), $$G_{i}^{1}$$ and $$G_{i}^{0}$$ denote the values of the objective functions $$G_{i}(x)$$ such that the degree of the membership function is 1 and 0, which can be determined by
\begin{aligned} G_{i}^{0}= \underset{x \in X}{\text {Min}}\, G_{i}(x)\qquad G_{i}^{1}= \underset{x \in X}{\text {Max}} \, G_{i}(x) \qquad \hbox { for } \quad i=1, 2, \ldots , p. \end{aligned}
Here, $$G_{i}(x)$$ is a concave function for $$i=1, 2, \ldots , p$$. The $${\text {Max}}_{x \in X}\, G_{i}(x)$$ is a convex programming problem and its optimal solution can be calculated without difficulty. For the problem $${\text {Min}}_{x \in X}\, G_{i}(x)$$, its optimal solution should be obtained at the boundary of the convex set X. If the problem $${\text {Min}}_{x \in X}\, G_{i}(x)$$ or $${\text {Max}}_{x \in X}\, G_{i}(x)$$ has no solution or $$G_{i}^{1}=\infty$$ or $$G_{i}^{0}=\infty$$, the decision maker may set the value of $$G_{i}^{1}$$, $$G_{i}^{0}$$ subjectively. Hence, Eq. (10) converted into the following form:
\begin{aligned} \text {Max }\,&\{\mu _{1}(G_{1}(x)), \mu _{2}(G_{2}(x)), \ldots , \mu _{p}(G_{p}(x))\} \\ \text {s. t. }&x \in X \end{aligned}
(12)
For each objective function $$\mu _{i}(G_{i}(x))$$, assume the decision maker gives the reference value of membership function $$\overline{\mu }_{i}$$ to reflect the desired value of membership function $$\mu _{i}(G_{i}(x))$$. Through solving the Min–Max Eq. (12), we obtain an efficient solution of Eq. (9) as follows:
\begin{aligned} \underset{i=1,2, \ldots , p}{\text {Min Max}}\,\,&[\overline{\mu }_{i} - \mu _{i}(G_{i}(x))] \\ \text {s. t. }&x \in X \end{aligned}
(13)
By proposing auxiliary variable $$\rho$$, Eq. (13) is equivalent to
\begin{aligned} \text {Min }\,&\rho \\ \text {s. t. }&\left\{ \begin{array}{ll} \overline{\mu }_{i} - \mu _{i}(G_{i}(x)) \le \rho \\ 0 \le \rho \le 1, \\ {\text {where}}\,\, x \in X, \quad i=1, 2, \ldots , p. \end{array}\right. \end{aligned}
(14)
or, equivalently Eq. (14) can be written by the following form
\begin{aligned} \text {Min }\,&\rho \\ \text {s. t. }&\left\{ \begin{array}{ll} b_{2}-2\alpha _{i}(b_{2}-b_{1})+\beta _{i}c_{ij}^{r}x \ge G_{i}^{0} + (\overline{\mu }_{i} - \rho )(G_{i}^{1}-G_{i}^{0}) \\ 0 \le \rho \le 1, \\ {\text {where}}\,\, x \in X, \quad i=1, 2, \ldots , p. \end{array}\right. \end{aligned}
(15)

## 4 Notation and assumptions

To develop the mathematical model of inventory replenishment intention, the notation affected in this paper is as below:

### 4.1 Notation

1. (i)

$$Q_{i}$$= the ordering quantity per cycle for $$i\hbox {th}$$ item

2. (ii)

$$A_{i}$$= the replenishment cost per order of $$i\hbox {th}$$ item

3. (iii)

$$c_{i}$$= purchasing cost of each product of the $$i\hbox {th}$$ item

4. (iv)

$$c_{1i}$$= shortage cost per unit time for $$i\hbox {th}$$ item

5. (v)

$$c_{3i}$$= the cost of lost sales per unit of $$i\hbox {th}$$ item

6. (vi)

$$S_{i}$$= shortage level for the $$i\hbox {th}$$ item

7. (vii)

$$D_{i}(t)$$=demand rate of $$i\hbox {th}$$ item, which is a function of inventory level at time t

8. (viii)

$$\vartheta$$=inventory level elasticity of demand rate($$0 \le \vartheta <1$$)

9. (ix)

$$t_{1i}$$= the time at which the inventory level reach zero for $$i\hbox {th}$$ item(a decision variable)

10. (x)

$$t_{2i}$$= the length of period during which are allowed for $$i\hbox {th}$$ item

11. (xi)

$$T_{i}$$= the length of the inventory cycle, hence $$T_{i}=t_{1i}+t_{2i}$$(a decision variable)

12. (xii)

$$H_{i}[q_{i}(t)]$$= holding cost for the $$i\hbox {th}$$ item, which is function of inventory level at time t

13. (xiii)

$$\gamma$$=holding cost elasticity($$\gamma \ge 1$$)

14. (xiv)

$$h_{i}$$= scaling constant for holding cost

15. (xv)

$$w_{i}$$ = storage space per unit quantity for the $$i\hbox {th}$$ item

16. (xvi)

B = budget available for replenishment

17. (xvii)

F = available storage space in the inventory system

In addition, the following assumptions are instated

### 4.2 Assumption

1. (i)

The replenishment rate is infinite and the lead-time zero.

2. (ii)

The time horizon of the inventory system is infinite.

3. (iii)

Shortage are allowed and during the stock-out period, a fraction $$\dfrac{1}{1+ \varepsilon _{i} x}$$ of the demand will be back order, and the remaining fraction $$(1-\dfrac{1}{1+ \varepsilon _{i} x})$$ will be lost, where x is the waiting time up to the next replenishment and $$\varepsilon _{i}$$ is a positive constant.

4. (iv)
The demand rate function $$D_{i}(t)$$ is deterministic and a power function of instantaneous stock level $$q_{i}(t)$$ at time t; that is:
\begin{aligned} D_{i}(t)=D_{i}[q_{i}(t)]= \left\{ \begin{array}{ll} \lambda _{i} [q_{i}(t)]^{\vartheta }, & \quad \text {if } 0 \le t \le t_{1i},\, q_{i}(t) \ge 0; \\ \lambda _{i}, & \quad \text {if } \, t_{1i} < t \le T_{i},\,q_{i}(t)> 0; \end{array} \right. \end{aligned}
where $$\lambda _{i}> 0$$ and $$0 \le \vartheta < 1$$.

5. (v)

the holding cost is non-linear function of the stock level $$q_{i}(t)$$ at time t and is given as $$H_{i}(t)=H_{i}[q_{i}(t)]=h_{i}[q_{i}(t)]^{\gamma }$$, where $$h_{i}> 0$$ and $$\gamma \ge 1$$.

## 5 Model formulation

Using the above assumption, the inventory level follows the pattern depicted in Fig. 2. The depletion of the inventory happens due to the effect of demand in $$[0, t_{1i})$$ and the demand backlogged in $$[t_{1i}, T_{i}]$$. Now, in the interval $$[0, T_{i}]$$, the inventory level gradually decreases to meet the demands. By this process, the inventory level reaches zero at $$t=t_{1i}$$, and shortages are allowed to occur in $$[t_{1i}, T_{i}]$$. Hence, the variety of inventory level $$q_{i}(t)$$, with respect to time t can be described by the following differential equations:
\begin{aligned} \dfrac{{\text {d}}q_{1i}(t)}{{\text {d}}t}= - \lambda _{i} [q_{i}(t)]^{\vartheta }, \quad 0 \le t \le t_{1i} \end{aligned}
(16)
with conditions $$q_{1i}(0)=R_{i}(=Q_{i}-S_{i})$$ and $$q_{1i}(t_{1i})=0$$
The solution of Eq. (16) is
\begin{aligned} q_{1i}(t)=\left[ R_{i}^{1-\vartheta }-(1-\vartheta ) \lambda _{i} t\right] ^{\dfrac{1}{1-\vartheta }} \end{aligned}
(17)
By the interval $$[t_{1i}, T_{i}]$$, the inventory level only build on demand, and some demand is lost while a fraction $$\dfrac{1}{1+\varepsilon _{i}(T_{i}-t)}$$ of the demand is backlogged, where $$t \in [t_{1i}, T_{i}]$$. In this case, the inventory level is controlled by the following differential equation:
\begin{aligned} \dfrac{{\text {d}}q_{2i}(t)}{{\text {d}}t}=-\dfrac{\lambda _{i}}{1+\varepsilon _{i}(T_{i}-t)}, \quad t_{1i} \le t \le T_{i} \end{aligned}
(18)
with conditions $$q_{2i}(T_{i})=-S_{i}$$ and $$q_{2i}(t_{1i})=0$$
The solution of Eq. (18) is
\begin{aligned} q_{2i}(t)=-\dfrac{\lambda _{i}}{\varepsilon _{i}} \left\{ \ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] -\ln \left[ 1+\varepsilon _{i}(T_{i}-t)\right] \right\} \end{aligned}
(19)
Considering the continuity of $$q_{1i}(t)$$ and $$q_{2i}(t)$$ at point $$t=t_{1i}$$, $$i.e., q_{1i}(t_{1i})=q_{2i}(t_{2i})=0$$, we have:
\begin{aligned} R_{i}=\left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }} \quad S _{i}=\dfrac{\lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \end{aligned}
(20)
Therefore, the ordering quantity $$(Q_{i})$$ over the replenishment cycle for the $$i\hbox {th}$$ item can be determined as
\begin{aligned} Q_{i}=q_{1i}(0)-q_{2i}(T_{i})=\left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}+\dfrac{\lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \end{aligned}
(21)
Based on Eqs. (17), (19) and (21), the total inventory cost per cycle consists of the following elements

The ordering cost per cycle for $$i\hbox {th}$$ item is $$A_{i}$$

The inventory holding cost per cycle for the $$i\hbox {th}$$ item is given by
\begin{aligned} h_{i}\int _{0}^{t_{1i}}[q_{1i}(t)]^{\gamma }{\text {d}}t&= h_{i} \int _{0}^{t_{1i}}\left[ R_{i}^{1-\vartheta } - (1- \vartheta )\lambda _{i} t\right] ^{\frac{\gamma }{1-\vartheta }} {\text {d}}t \\&= \dfrac{h_{i}}{(\gamma +1-\vartheta ) \lambda _{i}}R_{i}^{\gamma +1-\vartheta } \\&= \dfrac{h_{i}}{(\gamma +1-\vartheta ) \lambda _{i}} \left[ (1-\vartheta ) \lambda _{i} t_{1i}\right] ^{\frac{\gamma +1-\vartheta }{1-\vartheta }} \end{aligned}
(22)
Purchase cost of $$i\hbox {th}$$ item per cycle is
\begin{aligned} c_{i}Q_{i} = c_{i} \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}+\dfrac{c_{i} \lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \end{aligned}
(23)
The opportunity cost due to lost sales for $$i\hbox {th}$$ item is
\begin{aligned}&c_{3i} \lambda _{i} \int _{t_{1i}}^{T_{i}} \left( 1-\dfrac{1}{1+\varepsilon _{i}(T_{i}-t)}\right) {\text {d}}t \\&\quad = \dfrac{c_{3i} \lambda _{i}}{\varepsilon _{i}}\left\{ (T_{i}-t_{1i})-\ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] \right\} \end{aligned}
(24)
Sales revenue per cycle for the $$i\hbox {th}$$ item is given by
\begin{aligned}&s_{i} \int _{0}^{t_{1i}} D_{i}(t){\text {d}}t + s_{i}\int _{t_{1i}}^{T_{i}}\dfrac{D_{i}(t)}{[1+\varepsilon _{i}(T_{i}-t)]}{\text {d}}t\\&\quad = \int _{0}^{t_{1i}}s_{i}\lambda _{i}[q_{i}(t)]^{\vartheta } + \int _{t_{1i}}^{T_{i}}\dfrac{s_{i}\lambda _{i}}{[1+\varepsilon _{i}(T_{i}-t)]}{\text {d}}t \\&\quad = \dfrac{s_{i}\lambda _{i}}{\varepsilon _{i}}\left\{ \ln [1+\varepsilon _{i}(T_{i}-t_{1i})] + \dfrac{\varepsilon _{i}}{\lambda _{i}}[(1-\vartheta )\lambda _{i}t_{1i}]^{\frac{1}{1-\vartheta }}\right\} \end{aligned}
(25)
Shortage cost for the $$i\hbox {th}$$ item is given by
\begin{aligned}&c_{1i} \int _{t_{1i}}^{T_{i}}-q_{2i}(t){\text {d}}t \\&\quad =\dfrac{c_{1i}\lambda _{i}}{\varepsilon _{i}} \int _{t_{1i}}^{T_{i}} \left\{ \ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] - \ln \left[ 1+\varepsilon _{i}(T_{i}-t) \right] \right\} {\text {d}}t \\&\quad = \dfrac{c_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\left\{ (T_{i}-t_{1i})-\ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] \right\} \end{aligned}
(26)
In conjunction with the relevant inventory costs mentioned above, we can simplify the total profit (TP) and total wastage cost (WC) per inventory cycle as follows:
\begin{aligned} {\text {TP}}&= \dfrac{1}{T}[\text {sales revenue} - \text {ordering cost} - \text {holding cost} -\text {purchase cost}] \\ {\text {TP}}(t_{1i}, T_{i})&= \sum _{i=1}^{n}\dfrac{1}{T_{i}}\bigg [\dfrac{s_{i}\lambda _{i}}{\varepsilon _{i}}\left\{ \ln [1+\varepsilon _{i}(T_{i}-t_{1i})] + \dfrac{\varepsilon _{i}}{\lambda _{i}}[(1-\vartheta )\lambda _{i}t_{1i}]^{\frac{1}{1-\vartheta }}\right\} \\&\quad - \dfrac{h_{i}}{(\gamma +1-\vartheta ) \lambda _{i}}\left[ (1-\vartheta ) \lambda _{i} t_{1i}\right] ^{\frac{\gamma +1-\vartheta }{1-\vartheta }} \\&\quad - c_{i} \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}-\dfrac{c_{i} \lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} -A_{i} \bigg ] \\ {\text {WC}}&= \dfrac{1}{T}[\text {shortage cost + opportunity cost}] \end{aligned}
\begin{aligned} {\text {WC}}(t_{1i}, T_{i})= & \sum _{i=1}^{n}\dfrac{1}{T_{i}}\bigg [\left( \dfrac{c_{3i} \lambda _{i}}{\varepsilon _{i}}+\dfrac{c_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\right) (T_{i}-t_{1i}) \\&-\left( \dfrac{c_{3i} \lambda _{i}}{\varepsilon _{i}}+\dfrac{c_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\right) \ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] \bigg ] \end{aligned}
Our problem is to maximize the total profit and minimize the total wastage cost under two subjects to constraints, such as one budget constraint and another space constraint. Hence, the multi-objective multi-item crisp inventory problem is given by
\begin{aligned} \text {Max} \, {\text {TP}}(t_{1i}, T_{i})&=\sum _{i=1}^{n}\dfrac{1}{T_{i}}\bigg [\dfrac{s_{i}\lambda _{i}}{\varepsilon _{i}}\left\{ \ln [1+\varepsilon _{i}(T_{i}-t_{1i})] + \dfrac{\varepsilon _{i}}{\lambda _{i}}[(1-\vartheta )\lambda _{i}t_{1i}]^{\frac{1}{1-\vartheta }}\right\} -\dfrac{h_{i}}{(\gamma +1-\vartheta ) \lambda _{i}}\left[ (1-\vartheta ) \lambda _{i} t_{1i}\right] ^{\frac{\gamma +1-\vartheta }{1-\vartheta }} \\&\quad - c_{i} \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}-\dfrac{c_{i} \lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} -A_{i} \bigg ]\\ \text {Min}\, {\text {WC}}(t_{1i}, T_{i})&=\sum _{i=1}^{n}\dfrac{1}{T_{i}}\bigg [\left( \dfrac{c_{3i} \lambda _{i}}{\varepsilon _{i}}+\dfrac{c_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\right) (T_{i}-t_{1i})-\left( \dfrac{c_{3i} \lambda _{i}}{\varepsilon _{i}}+\dfrac{c_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\right) \ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] \bigg ] \\ \text {subject to }&\sum _{i=1}^{n}c_{i}\left[ \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}+\dfrac{\lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \right] \le B; \\&\sum _{i=1}^{n}w_{i}\left[ \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}+\dfrac{\lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \right] \le F;\\&t_{1}\ge 0,\ T\ge 0 \end{aligned}
(27)
where $$t_{1}=(t_{11}, t_{12}, \ldots , t_{1n})^{\mathbf{T}}$$ and $$T=(T_{1}, T_{2}, \ldots , T_{n})^{\mathbf{T}}$$ are decision variables.

### 5.1 Fuzzy-rough inventory model

When $$c_{i}, c_{1i}, c_{3i}, h_{i}, s_{i}, w_{i}, A_{i}, B$$ and F become fuzzy rough variables, the proposed multi-objective crisp inventory problem shown in Eq. (27), which can be formulated by the following model:
\begin{aligned} \text {Max}\, \widehat{\tilde{{\text {TP}}}}(t_{1i}, T_{i})&=\sum _{i=1}^{n}\dfrac{1}{T_{i}}\bigg [\dfrac{\widehat{\tilde{s}}_{i} \lambda _{i}}{\varepsilon _{i}}\left\{ \ln [1+\varepsilon _{i}(T_{i}-t_{1i})] + \dfrac{\varepsilon _{i}}{\lambda _{i}}[(1-\vartheta )\lambda _{i}t_{1i}]^{\frac{1}{1-\vartheta }}\right\} -\dfrac{\widehat{\tilde{h}}_{i}}{(\gamma +1-\vartheta ) \lambda _{i}}\left[ (1-\vartheta ) \lambda _{i} t_{1i}\right] ^{\frac{\gamma +1-\vartheta }{1-\vartheta }} \\&\qquad - \widehat{\tilde{c}}_{i} \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}-\dfrac{\widehat{\tilde{c}}_{i} \lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} -\widehat{\tilde{A}}_{i} \bigg ] \\ \text {Min}\, \widehat{\tilde{{\text {WC}}}}(t_{1i}, T_{i})&=\sum _{i=1}^{n}\dfrac{1}{T_{i}}\bigg [\left( \dfrac{\widehat{\tilde{c}}_{3i} \lambda _{i}}{\varepsilon _{i}}+\dfrac{\widehat{\tilde{c}}_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\right) (T_{i}-t_{1i})-\left( \dfrac{\widehat{\tilde{c}}_{3i} \lambda _{i}}{\varepsilon _{i}}+\dfrac{\widehat{\tilde{c}}_{1i} \lambda _{i}}{\varepsilon _{i}^{2}}\right) \ln \left[ 1+\varepsilon _{i}(T_{i}-t_{1i})\right] \bigg ] \\ \text {s. t. }&\widehat{\tilde{{\text {BC}}}}=\sum _{i=1}^{n}\widehat{\tilde{c}}_{i}\left[ \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}+\dfrac{\lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \right] \le \widehat{\tilde{B}}; \\&\widehat{\tilde{{\text {SC}}}}=\sum _{i=1}^{n}\widehat{\tilde{w}}_{i}\left[ \left[ (1-\vartheta )\lambda _{i} t_{1i}\right] ^{\frac{1}{1-\vartheta }}+\dfrac{\lambda _{i}}{\varepsilon _{i}} \ln \left\{ 1+\varepsilon _{i}(T_{i}-t_{1i})\right\} \right] \le \widehat{\tilde{F}}; \\&t_{1}\ge 0,\ T\ge 0 \end{aligned}
where $$t_{1}=(t_{11}, t_{12}, \ldots , t_{1n})^{\mathbf{T}}$$ and $$T=(T_{1}, T_{2}, \ldots , T_{n})^{\mathbf{T}}$$ are decision variables.
Or, equivalently
\begin{aligned} \begin{array}{ll} {\text {Max}} & \quad \widehat{\tilde{Z}}_{1}(\eta ) = \widehat{\tilde{TP}}(t_{1i}, T_{i}) \\ {\text {Min}} & \quad \widehat{\tilde{Z}}_{2} (\eta ) = \widehat{\tilde{WC}}(t_{1i}, T_{i}) \\ \text {s.t.} & \quad \widehat{\tilde{{\text {BC}}}}(\eta ) \le \widehat{\tilde{B}} \\ & \quad \widehat{\tilde{{\text {SC}}}} (\eta ) \le \widehat{\tilde{F}} \\ & \quad t_{1}\ge 0,\ T\ge 0, \end{array} \end{aligned}
(28)
where $$t_{1}=(t_{11}, t_{12}, \ldots , t_{1n})^{\mathbf{T}}$$ and $$T=(T_{1}, T_{2}, \ldots , T_{n})^{\mathbf{T}}$$ are decision variables.

Therefore, the proposed fuzzy rough multi-objective problem shown in Eq. (28), which is the fuzzy representation form of the proposed crisp model Eq. (27).

### 5.2 Solution methodology

To solve the proposed fuzzy rough multi-objective multi-item inventory model (Eq. 28). We transformed the fuzzy rough model (Eq. 28) into its fuzzy rough Tr-Pos constrained the multi-objective model. Thus, we have the following fuzzy rough Tr-Pos constrained multi-objective model:
\begin{aligned} \text {Max }\,&\{f_{1}, f_{2}\} \\ \text {s. t. }&\left\{ \begin{array}{ll} Tr\{\eta | Pos\{\widehat{\tilde{Z}}_{1}(\eta ) \ge f_{1}\} \ge \beta _{1}\} \ge \alpha _{1}, \\ Tr\{\eta | Pos\{- \widehat{\tilde{Z}}_{2}(\eta ) \ge f_{2}\} \ge \beta _{2}\} \ge \alpha _{2}, \\ Tr\{\eta | Pos\{\widehat{\tilde{{\text {BC}}}}(\eta ) \le \widehat{\tilde{B}}\} \ge \sigma _{1}\} \ge \delta _{1}, \\ Tr\{\eta | Pos\{\widehat{\tilde{{\text {SC}}}}(\eta ) \le \widehat{\tilde{F}}\} \ge \sigma _{2}\} \ge \delta _{2}, \\ t_{1}\ge 0,\ T\ge 0. \end{array}\right. \end{aligned}
(29)
In Eq. (29), $$\widehat{\tilde{Z}}_{1}, \widehat{\tilde{Z}}_{2}, \widehat{\tilde{{\text {BC}}}}$$ and $$\widehat{\tilde{{\text {SC}}}}$$ all are fuzzy variables. Therefore, we considered that the fuzzy variables $$\widehat{\tilde{Z}}_{1}, \widehat{\tilde{Z}}_{2}, \widehat{\tilde{{\text {BC}}}}$$ and $$\widehat{\tilde{{\text {SC}}}}$$ are characterized by the following membership functions:
\begin{aligned} \mu _{\widehat{\tilde{Z}}_{1}(\eta )}(t) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{Z}_{1}^{1}}{\widehat{Z}_{1}^{2}-\widehat{Z}_{1}^{1}} & \,\,{\text {if}}\,\, \widehat{Z}_{1}^{1}\le t \le \widehat{Z}_{1}^{2} \\ 1 & \,\,{\text {if}}\,\, t=\widehat{Z}_{1}^{2} \\ \dfrac{\widehat{Z}_{1}^{3}-t}{\widehat{Z}_{1}^{3}-\widehat{Z}_{1}^{2}} & \,\,{\text {if}}\,\, \widehat{Z}_{1}^{2} \le t \le \widehat{Z}_{1}^{3} \\ 0 & \,{\text {otherwise}} \end{array}\right. \end{aligned}
\begin{aligned} \mu _{\widehat{\tilde{Z}}_{2}(\eta )}(t) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{Z}_{2}^{1}}{\widehat{Z}_{2}^{2}-\widehat{Z}_{2}^{1}} & \,\,{\text {if}}\,\, \widehat{Z}_{1}^{1}\le t \le \widehat{Z}_{1}^{2} \\ 1 & \,\,{\text {if}}\,\, t=\widehat{Z}_{2}^{2} \\ \dfrac{\widehat{Z}_{2}^{3}-t}{\widehat{Z}_{2}^{3}-\widehat{Z}_{2}^{2}} & \,\,{\text {if}}\,\, \widehat{Z}_{2}^{2} \le t \le \widehat{Z}_{2}^{3} \\ 0 & \,{\text {otherwise}} \end{array}\right. \end{aligned}
and
\begin{aligned} \mu _{\widehat{\tilde{{\text {BC}}}}(\eta )}(t) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{{\text {BC}}}^{1}}{\widehat{{\text {BC}}}^{2}-\widehat{{\text {BC}}}^{1}} & \,\,{\text {if}}\,\, \widehat{{\text {BC}}}^{1}\le t \le \widehat{{\text {BC}}}^{2} \\ 1 & \,\,{\text {if}}\,\, t=\widehat{{\text {BC}}}^{2} \\ \dfrac{\widehat{{\text {BC}}}^{3}-t}{\widehat{{\text {BC}}}^{3}-\widehat{{\text {BC}}}_{1}^{2}} & \,\,{\text {if}}\,\, \widehat{{\text {BC}}}^{2} \le t \le \widehat{{\text {BC}}}^{3} \\ 0 & \,{\text {otherwise}} \end{array}\right. \end{aligned}
\begin{aligned} \mu _{\widehat{\tilde{{\text {SC}}}}(\eta )}(t) = \left\{ \begin{array}{ll} \dfrac{t-\widehat{{\text {SC}}}^{1}}{\widehat{{\text {SC}}}^{2}-\widehat{{\text {SC}}}^{1}} & \,\,{\text {if}}\,\, \widehat{{\text {SC}}}^{1}\le t \le \widehat{{\text {SC}}}^{2} \\ 1 & \,\,{\text {if}}\,\, t=\widehat{{\text {SC}}}^{2} \\ \dfrac{\widehat{{\text {SC}}}^{3}-t}{\widehat{{\text {SC}}}^{3}-\widehat{{\text {SC}}}^{2}} & \,\,{\text {if}}\,\, \widehat{{\text {SC}}}^{2} \le t \le \widehat{{\text {SC}}}^{3} \\ 0 & \,{\text {otherwise}} \end{array}\right. \end{aligned}
Here, we assume that $$\widehat{Z}_{1}=([a_{11}, a_{21}][b_{11}, b_{21}])$$, $$\widehat{Z}_{2}=([a_{12}, a_{22}][b_{12}, b_{22}])$$, $$\widehat{{\text {BC}}}=([a_{13}, a_{23}][b_{13}, b_{23}])$$ and $$\widehat{{\text {SC}}}=([a_{14}, a_{24}][b_{14}, b_{24}])$$ are rough variables respectively. For the cases: $$a_{21} \le f_{1} -\beta _{1}Z_{1}^{r}T \le b_{21}$$, $$a_{22} \le f_{2} -\beta _{2}Z_{2}^{r}T \le b_{22}$$, $$a_{23} \le \sigma _{1} (BC)^{l}T + \sigma _{1} B^{r} \le b_{23}$$ and $$a_{24} \le \sigma _{2} (SC)^{l}T + \sigma _{2} F^{r} \le b_{24}$$, by the Theorems 3.1 & 3.2, we get the following deterministic problem of the proposed model (29).
\begin{aligned} \text {Max }\,&G_{1}(T) = b_{21}-2\alpha _{1}(b_{21}-b_{11})+\beta _{1}Z_{1}^{r}T \\ \text {Max }\,&G_{2}(T) = b_{22}-2\alpha _{2}(b_{22}-b_{12})+\beta _{2}Z_{2}^{r}T \\ \text {s. t. }&\left\{ \begin{array}{ll} \sigma _{1} (BC)^{l}T + \sigma _{1} B^{r} \ge (2\delta _{1} - 1)(b_{23} - b_{13}) + b_{13}, \\ \sigma _{2} (SC)^{l}T + \sigma _{2} F^{r} \ge (2\delta _{2} - 1)(b_{24} - b_{14}) + b_{14}, \\ T \ge 0, \delta _{1}, \sigma _{1} \in [0,1], \\ \alpha _{1}, \alpha _{2}, \beta _{1}, \beta _{2} \in [0, 1] \end{array}\right. \end{aligned}
(30)
where, $$Z_{1}^{r}, Z_{2}^{r}, B^{r}, F^{r}$$ are positive numbers expressing the right spread of $$\widehat{\tilde{Z}}_{1}, \widehat{\tilde{Z}}_{2}, \widehat{\tilde{B}}, \widehat{\tilde{F}}$$ and $$(BC)^{l}, (SC)^{l}$$ are positive numbers expressing the left spread of $$\widehat{\tilde{{\text {BC}}}}, \widehat{\tilde{{\text {SC}}}}$$, respectively. For the model (29), by Eq. (30) in the interactive fuzzy satisfying method, we get
\begin{aligned} \text {Min }\,&\rho \\ \text {s. t. }&\left\{ \begin{array}{ll} b_{21}-2\alpha _{1}(b_{21}-b_{11})+\beta _{1}Z_{1}^{r}T \ge G_{1}^{0} + (\overline{\mu }_{1} - \rho )(G_{1}^{1}-G_{1}^{0}), \\ b_{22}-2\alpha _{1}(b_{22}-b_{12})+\beta _{2}Z_{2}^{r}T \ge G_{2}^{0} + (\overline{\mu }_{2} - \rho )(G_{2}^{1}-G_{2}^{0}), \\ \sigma _{1} (BC)^{l}T + \sigma _{1} B^{r} \ge (2\delta _{1} - 1)(b_{23} - b_{13}) + b_{13}, \\ \sigma _{2} (SC)^{l}T + \sigma _{2} F^{r} \ge (2\delta _{2} - 1)(b_{24} - b_{14}) + b_{14}, \\ T \ge 0, \delta _{1}, \sigma _{1} \in [0,1], \\ \alpha _{1}, \alpha _{2}, \beta _{1}, \beta _{2} \in [0, 1] \end{array}\right. \end{aligned}
(31)

## 6 Numerical example

In this section, we present a numerical example to illustrate the proposed model. CFL is a famous manufacturer company of electronic products, a leader in equipment and services in India. With the development and innovation of technology, CFL company will produce three new products which can be classified into high-end, mid-end, and low-end types of products according to different customer in order to increase market share. The quality and credibility of products and services are among the most important factors driving customer satisfaction and fidelity. Quality management is essential for leveraging newness globally and improving productivity in general. Here, we assume that the managers of the company are interested in three new items I (high-end), II (mid-end), III (low-end), and want these new items, which were manufactured by its own producers, to be sold around the whole country.

When one reputed company launches a new product on the market, they cannot decreases the reputation level in the market. Therefore, the defective electronic items will be produced in the process of reproduction. Since these items are never been launched on the market. Therefore, here many uncertain factor is working. Due to many uncertain factors, we assume that the total profit (TP), wastage cost (WC), sales revenue cost, purchasing cost, selling price, ordering cost and shortage cost of the inventory problem are fuzzy rough variables.

Other assumed conditions are as follows: the available storage area is $$\widehat{\tilde{F}}=(\widehat{F}-50, \widehat{F}, \widehat{F}+70)$$ square meters(where $$\widehat{F} \in ([150, 180] [160, 210])$$) and the available budget is $$\widehat{\tilde{B}}=\(\widehat{B}-120, \widehat{B}, \widehat{B}+160)$$(where $$\widehat{B} \in ([580, 600] [560, 680])$$). On the basis of former experience, the company is interested in minimizing the wastage cost(WC) and maximize the total profit(TP).

By the limitations of both storage space and total budget cost, the formulation of above fuzzy-rough multi-objective inventory problem is shown in (28). Then the proposed fuzzy rough multi-objective inventory problem is solved under Tr-Pos technique. The fuzzy rough cost parameters for the proposed model are presented in Table 2, and other crisp parameter values for three items (n = 3) are given in Table 3. Finally, Tables 4, 5 and 6 shows the different optimal results using various confidence levels: $$\alpha _{i}=0.9, \delta _{j}= 0.9$$-Tr, $$\beta _{i}=0.9, \sigma _{j}=0.9$$-Pos.
Table 2

Input fuzzy rough cost parameters

Item

I

II

III

$$\widehat{\tilde{s}}_{i}$$

$$(\widehat{s}_{1}-8.0, \widehat{s}_{1}, \widehat{s}_{1}+10)$$

$$(\widehat{s}_{2}-7.0, \widehat{s}_{2}, \widehat{s}_{2}+10)$$

$$(\widehat{s}_{3}-5.0, \widehat{s}_{3}, \widehat{s}_{3}+8.0)$$

$$\widehat{s}_{1} \in ([32, 38] [31, 40])$$

$$\widehat{s}_{2} \in ([36, 40] [34, 42])$$

$$\widehat{s}_{3} \in ([35, 36] [31, 40])$$

$$\widehat{\tilde{h}}_{i}$$

$$(\widehat{h}_{1}-2.0, \widehat{h}_{1}, \widehat{h}_{1}+2.0)$$

$$(\widehat{h}_{2}-1.5, \widehat{h}_{2}, \widehat{h}_{2}+3.0)$$

$$(\widehat{h}_{3}-2.0, \widehat{h}_{3}, \widehat{h}_{3}+1.0)$$

$$\widehat{h}_{1} \in ([6.0, 7.5] [5.0, 8.0])$$

$$\widehat{h}_{2} \in ([7.0, 8.0] [6.0, 9.0])$$

$$\widehat{h}_{3} \in ([8.0, 9.5] [7.0, 10.0])$$

$$\widehat{\tilde{c}}_{i}$$

$$(\widehat{c}_{1}-1.0, \widehat{c}_{1},\widehat{c}_{1}+2.0)$$

$$(\widehat{c}_{2}-1.5, \widehat{c}_{2}, \widehat{c}_{2}+1.0)$$

$$(\widehat{c}_{3}-2.0, \widehat{c}_{3}, \widehat{c}_{3}+3.0)$$

$$\widehat{c}_{1} \in ([12, 14] [11, 15])$$

$$\widehat{c}_{2} \in ([16, 18] [15, 19])$$

$$\widehat{c}_{3} \in ([12, 14] [10, 15])$$

$$\widehat{\tilde{c}}_{1i}$$

$$(\widehat{c}_{11}-3.0, \widehat{c}_{11}, \widehat{c}_{11}+3.5)$$

$$(\widehat{c}_{12}-2.0, \widehat{c}_{12}, \widehat{c}_{12}+4.0)$$

$$(\widehat{c}_{13}-1.0, \widehat{c}_{13}, \widehat{c}_{13}+3.0)$$

$$\widehat{c}_{11} \in ([10, 12] [9.0, 16])$$

$$\widehat{c}_{12} \in ([13, 14] [12, 18])$$

$$\widehat{c}_{13} \in ([9.0, 11] [8.0, 17])$$

$$\widehat{\tilde{c}}_{3i}$$

$$(\widehat{c}_{31}-10, \widehat{c}_{31}, \widehat{c}_{31}+13)$$

$$(\widehat{c}_{32}-12, \widehat{c}_{32}, \widehat{c}_{32}+14)$$

$$(\widehat{c}_{33}-8.0, \widehat{c}_{33}, \widehat{c}_{33}+12)$$

$$\widehat{c}_{31} \in ([40, 42] [38, 52])$$

$$\widehat{c}_{32} \in ([43, 45] [40, 58])$$

$$\widehat{c}_{33} \in ([35, 37] [34, 46])$$

$$\widehat{\tilde{w}}_{i}$$

$$(\widehat{w}_{1}-1.0, \widehat{w}_{1}, \widehat{w}_{1}+1.0)$$

$$(\widehat{w}_{2}-1.0, \widehat{w}_{2}, \widehat{w}_{2}+1.5)$$

$$(\widehat{w}_{3}-1.5, \widehat{w}_{3}, \widehat{w}_{3}+2.0)$$

$$\widehat{w}_{1} \in ([3.0, 3.5] [2.0, 4.0])$$

$$\widehat{w}_{2} \in ([4.0, 4.5] [3.0, 5.0])$$

$$\widehat{w}_{3} \in ([4.0, 5.0] [3.5, 6.0])$$

$$\widehat{\tilde{A}}_{i}$$

$$(\widehat{A}_{1}-8.0, \widehat{s}_{1}, \widehat{A}_{1}+8.0)$$

$$(\widehat{A}_{2}-5.0, \widehat{A}_{2}, \widehat{A}_{2}+7.0)$$

$$(\widehat{A}_{3}-8.0, \widehat{A}_{3}, \widehat{A}_{3}+12)$$

$$\widehat{A}_{1} \in ([40, 80] [30, 90])$$

$$\widehat{A}_{2} \in ([45, 85] [25, 95])$$

$$\widehat{A}_{3} \in ([50, 90] [25, 100])$$

Table 3

Input crisp parameters

Item

I

II

III

$$\lambda _{i}$$

9.50

9.00

10.5

$$\varepsilon _{i}$$

0.80

0.70

0.75

Table 4 shows different optimal solutions for fixed values of $$\gamma$$, $$\vartheta$$ and various value of $$\mu _{1}, \mu _{2}$$.

In Table 5, we have shown that different optimal solutions for a fixed value of $$\gamma$$ and different values of $$\vartheta$$. Here, for a fixed value of $$\gamma$$, the values of total profit(TP) is increased and wastage cost(WC) is decreasing as the value of $$\vartheta$$ increases. From an economic viewpoint, when the elasticity of the demand rate is high, the CFL company manager will maintain a high level of initial and ending inventories to increase the demand rate.
In Table 6, we have shown that different optimal solutions for a fixed value of $$\vartheta$$ and different values of $$\gamma$$. Here, for a fixed value of $$\vartheta$$, the values of total profit(TP) is decreasing and wastage cost(WC) is increasing as the value of $$\gamma$$ increases. Hence, the CFL company lowers the ending inventory as far as possible or even permits shortages.
Table 4

Optimum result for fixed value of $$\vartheta = 0.05$$ and $$\gamma = 1.30$$

$$\mu _{1}$$

$$\mu _{2}$$

$$t_{1i}$$

$$t_{2i}$$

$$T_{i}$$

$$Q_{i}$$

$$\rho$$

Max TP

Min WC

1

1

1.2626

0.2161

1.4787

14.845

0.1156

309.518

13.995

1.1585

0.2626

1.4211

13.346

1.0577

0.2532

1.3110

14.379

1

0.98

1.2589

0.2304

1.4894

14.921

0.1090

311.819

15.619

1.1542

0.2797

1.4339

13.431

1.0538

0.2690

1.3229

14.471

1

0.96

1.2553

0.2446

1.5000

14.996

0.1028

313.997

17.286

1.1499

0.2967

1.4467

13.515

1.0501

0.2845

1.3347

14.561

1

0.94

1.2518

0.2587

1.5106

15.069

0.0967

316.059

18.993

1.1458

0.3136

1.4594

13.598

1.0464

0.2999

1.3463

14.650

1

0.92

1.2483

0.2727

1.5211

15.142

0.0913

318.015

20.737

1.1417

0.3303

1.4720

13.680

1.0428

0.3150

1.3578

14.736

0.98

1

1.2663

0.2016

1.4680

14.768

0.1026

307.082

12.417

1.1630

0.2452

1.4082

13.258

1.0618

0.2372

1.2990

14.284

0.96

1

1.2702

0.1870

1.4572

14.687

0.0900

304.499

10.890

1.1675

0.2276

1.3952

13.169

1.0659

0.2208

1.2868

14.187

0.94

1

1.2741

0.1722

1.4463

14.609

0.0778

301.756

9.418

1.1723

0.2098

1.3821

13.078

1.0702

0.3042

1.2744

14.087

0.92

1

1.2781

0.1572

1.4353

14.526

0.0618

298.834

8.008

1.1771

0.1917

1.3689

12.984

1.0747

0.1871

1.2618

13.984

Table 5

Optimum result for fixed value of $$\mu _{1}=1, \mu _{2}=0.95$$ and $$\gamma = 1.30$$

$$\vartheta$$

$$t_{1i}$$

$$t_{2i}$$

$$T_{i}$$

$$Q_{i}$$

$$\rho$$

Max TP

Min WC

0.05

1.2535

0.2517

1.5053

15.033

0.0998

315.042

18.135

1.1478

0.3052

1.4530

13.557

1.0482

0.2922

1.3405

14.606

0.06

1.2636

0.2394

1.5030

15.252

0.0874

319.391

16.631

1.1580

0.2917

1.4498

13.723

1.0586

0.2788

1.3375

14.796

0.07

1.2738

0.2262

1.5000

15.479

0.0746

323.886

15.077

1.1684

0.2772

1.4457

13.892

1.0691

0.2644

1.3336

14.989

0.08

1.2740

0.2114

1.4854

15.587

0.0613

328.512

13.478

1.1682

0.2601

1.4284

13.934

1.0712

0.2479

1.3192

15.068

0.09

1.2686

0.1953

1.4640

15.623

0.0479

333.223

11.849

1.1621

0.2413

1.4035

13.900

1.0685

0.2299

1.2985

15.075

0.10

1.2638

0.1783

1.4421

15.660

0.0342

338.006

10.196

1.5566

0.2214

1.3781

13.866

1.0664

0.2108

1.2773

15.082

0.11

1.2595

0.1602

1.4197

15.698

0.0204

342.844

8.523

1.1521

0.2002

1.3523

13.832

1.0648

0.1905

1.2554

15.089

Table 6

Optimum result for fixed values of $$\mu _{1}=1, \mu _{2}=0.95$$ and $$\vartheta = 0.05$$

$$\gamma$$

$$t_{1i}$$

$$t_{2i}$$

$$T_{i}$$

$$Q_{i}$$

$$\rho$$

Max TP

Min WC

1.15

1.3863

0.1671

1.5535

15.781

0.0195

343.159

8.414

1.2410

0.2047

1.4458

13.737

1.1619

0.1982

1.3602

15.126

1.20

1.3541

0.1986

1.5528

15.693

0.0451

334.204

11.510

1.2215

0.2420

1.4636

13.829

1.1328

0.2334

1.3662

15.096

1.25

1.3230

0.2278

1.5509

15.593

0.0721

324.762

14.775

1.2048

0.2774

1.4822

13.928

1.1062

0.2661

1.3723

15.068

1.30

1.2535

0.2517

1.5053

15.033

0.0998

315.042

18.135

1.1478

0.3052

1.4530

13.557

1.0482

0.2922

1.3405

14.606

1.35

1.1688

0.2700

1.4389

14.264

0.1265

305.709

21.361

1.0727

0.3250

1.3977

12.942

0.9774

0.3117

1.2892

13.933

1.40

1.0942

0.2855

1.3797

13.583

0.1518

296.840

24.427

1.0066

0.3412

1.3479

12.393

0.9153

0.3278

1.2431

13.335

1.45

1.0281

0.2986

1.3268

12.978

0.1759

288.429

27.335

0.9481

0.3548

1.3029

11.902

0.8603

0.3414

1.2018

12.801

## 7 Discussion

We now studied the effects of changes in the values of the system parameters $$\vartheta , \gamma$$ and membership function values ($$\mu _{i}$$) on the total profit and wastage cost, respectively. The sensitivity analysis is performed by changing each of the parameters, taking one parameter at a time and keeping the remaining parameters unvaried. Analysis is based on the proposed example, and the results of the analysis are shown in Tables 4, 5 and 6. From Tables 4, 5, and 6, the following points are constructed.
1. (i)

For fixed value of $$\mu _{1}(=1)$$, the both value of total profit (TP) and wastage cost (WC) increase as the value $$\mu _{2}$$ decreases. Similarly, for fixed value of $$\mu _{2}(=1)$$, the both value of total profit (TP) and wastage cost (WC) decrease as the value $$\mu _{1}$$ decreases (cf. Figs. 3, 4).

2. (ii)

For fixed value of $$\gamma (>0)$$, the total profit(TP) increase and wastage cost(WC) decrease as the value $$\vartheta$$ increases (cf. Figs. 7, 11).

3. (iii)

For fixed value of $$\vartheta (>0)$$, the total profit(TP) decrease and wastage cost(WC) increase as the value $$\gamma$$ increases (cf. Figs. 8, 12).

4. (iv)

For fixed value of $$\gamma (>0)$$, the length of inventory cycle($$T_{i}$$) decrease while order quantity($$Q_{i}$$) increase with the increase in the value of the parameter $$\vartheta$$ (cf. Figs. 5, 9).

5. (v)

For fixed value of $$\vartheta (>0)$$, the order quantity($$Q_{i}$$) decrease and length of inventory cycle($$T_{i}$$) decrease with the increases the value of parameter $$\gamma$$ (cf. Figs. 6, 10).

Here, we have considered fuzzy rough multi-objective inventory problem. This fuzzy rough multi-objective inventory problem is converted to a deterministic problem using Tr-Pos constrained method. Now, the deterministic inventory problem is solved by three different methods which are fuzzy interactive satisfied method (FISM), global criteria method (GCM) and convex combination method (CCM) Chakraborty et al. (2014). The optimum result of these three methods are given in Table 7. From Table 7, it is clear that by the FISM we obtain satisfactory solutions of the proposed problem. So, the proposed methodology is a very useful tool in this kind of inventory problem (Figs. 5, 6, 7, 8, 9, 10, 11, 12).
Table 7

Optimum result for fixed values of $$\vartheta =0.05$$ and $$\gamma = 1.15$$ by different methods

Methods

$$t_{1i}$$

$$t_{2i}$$

$$T_{i}$$

$$Q_{i}$$

Max TP

Min WC

FISM

1.3863

0.1671

1.5535

15.781

343.159

8.414

1.2410

0.2047

1.4458

13.737

1.1619

0.1982

1.3602

15.126

GCM

1.2726

0.2324

1.5050

14.052

329.391

18.631

1.2852

0.2272

1.5124

12.327

1.3423

0.3642

1.7065

14.096

CCM

1.3728

0.2952

1.6680

13.974

323.685

20.451

1.3184

0.3672

1.6856

11.563

1.2591

0.4644

1.7235

14.707

## 8 Conclusion

In this paper, we investigated a multi-objective multi-item inventory model under both stock-dependent demand rate and holding cost rate with relaxed terminal conditions. To capture the real life business situations, different types of cost and other parameters of the proposed inventory model are considered in fuzzy rough environments. Tr-Pos chance constrained approaches have been proposed to solve this kind of inventory problem. Here, the interactive fuzzy satisfying method is exercised to solve a special type of fuzzy rough multi-objective inventory problem. Furthermore, we presented a numerical example to demonstrate our proposed methodology. This numerical example discloses that (1) total profit (TP) increase while the wastage cost (WC) decrease with an increase in a value of the parameter ‘$$\vartheta$$’ (cf. Table 5), (2) total profit (TP) decrease while wastage cost (WC) increase with the increase in a value of the holding cost elasticity ‘$$\gamma$$’ (cf. Table 6). However, the proposed model can be further extended in several ways like fuzzy demand, trapezoidal type demand, variable rate reworking and quantity discount, time-dependent holding cost and others.

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## Authors and Affiliations

• Totan Garai
• 1
• Dipankar Chakraborty
• 1
• Tapan Kumar Roy
• 1
1. 1.Department of MathematicsIndian Institute of Engineering Science and Technology, ShibpurHowrahIndia