A fuzzy rough multiobjective multiitem inventory model with both stockdependent demand and holding cost rate
Abstract
In this paper, we developed a multiobjective multiitem inventory model with fuzzy rough coefficients. Here, we have considered both demand and holding cost which is a nonlinear function of the instantaneous stock level. Chanceconstrained fuzzy rough multiobjective 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 TrPos chance constrained technique to determine the optimal solutions of a fuzzy rough multiobjective 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
Multiobjective Multiitem inventory Stockdependent demand Stockdependent holding cost Fuzzy rough variable Chance measure1 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 decisionmaking 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, setup 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, setup 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 fuzzyrough environment.
For the inventory problem, the classical inventory decisionmaking models have deliberated a single item. However, single item inventories rarely occur, whereas multiitem inventories are common in reallife circumstances. Many researchers (cf. Balkhi and Foul 2009; Hartley 1978; Lee and Yao 1998; Taleizadeh et al. 2011) investigated the multiitem 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 multiitem inventory model with stockdependent consumption rates. A replenishment model for noninstantaneous deteriorating items with stockdependent 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).
Summary of related literature for inventory model
Methods  Item  Objective  Demand rate  Holding cost  Shortage  Environment 

Pando et al. (2012)  Single  Single  Stockdependent (power function)  Nonlinearstock dependent  No  Crisp 
Taleizadeh et al. (2013)  Single  Single  Constant  Constant  Completely backlogged  Crisp 
Tripathi (2013)  Single  Single  Timedependent (power function)  Lineartime dependent  No  Crisp 
Xu and Zaho (2008)  Multi  Multi  Constant  Constant  No  Fuzzy rough 
Mondal et al. (2013b)  Multi  Single  Stockdependent (linear function)  Constant  No  Fuzzy rough 
Xu and Zhao (2010)  Multi  Multi  Constant  Constant  No  Fuzzy rough 
Pando et al. (2013)  Single  Single  Stockdependent (power function)  Nonlinear stock and time dependent  No  Crisp 
Min et al. (2012)  Single  Single  Stockdependent (power function)  Constant  No  Crisp 
Yang (2014)  Single  Single  Stockdependent (power function)  Nonlinearstock dependent  Partially backlogged  Crisp 
Pando et al. (2012)  Multi  Single  Pricedependent (power function)  Lineartime dependent  No  Crisp 
This proposed method  Multi  Multi  Stockdependent (power function)  Nonlinearstock dependent  Partially backlogged  Fuzzy rough 

TrPos constrained multiobjective model with fuzzy rough variables.

Multiobjective multiitem inventory model with demand is a power function and holding cost is a nonlinear function of the stock level.

Multiobjective multiitem inventory model under fuzzy rough environments.
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 nonempty 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
Trust theory is a branch of mathematics that studies the behaviour of rough event.
Definition 2.3
 (i)
\(\pi \{\Lambda \} < +\infty\);
 (ii)
\(\pi \{\Delta \}> 0\);
 (iii)
\(\pi \{A\}>0\) for any \(A \in \mathscr {A}\);
 (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
Definition 2.5
When we do not have enough information to determine the measure \(\pi\) for a reallife 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.
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
Definition 2.7
(Xu and Zhao 2010) An ndimensional fuzzy rough vector is a function \(\zeta\) from a rough space \((\Lambda , \Delta , \mathscr {A}, \pi )\) to the set of ndimensional 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 ntuple 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 TrPos constrained multiobjective model with fuzzy rough variable
Fuzzy programming of the multiobjective problem has been well manifested. It has been increasingly acknowledged that many realworld decisionmaking problems involve multiple and competing objectives which should be deliberated together. As a propagation of the fuzzy multiobjective decisionmaking case, the fuzzy rough multiobjective linear decisionmaking model (Xu and Zhao 2010) is defined as a means for optimizing multiple several objective functions subject to a number of constraints.
Proposition 2
Theorem 3.1
Proof
Theorem 3.2
Proof
Proof is similar as Theorem 3.1. \(\square\)
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 multiobjective programming Eq. (8) as our research objective and use the interactive fuzzy satisfying method to get an optimal solution to Eq. (5).
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
 (i)
\(Q_{i}\)= the ordering quantity per cycle for \(i\hbox {th}\) item
 (ii)
\(A_{i}\)= the replenishment cost per order of \(i\hbox {th}\) item
 (iii)
\(c_{i}\)= purchasing cost of each product of the \(i\hbox {th}\) item
 (iv)
\(c_{1i}\)= shortage cost per unit time for \(i\hbox {th}\) item
 (v)
\(c_{3i}\)= the cost of lost sales per unit of \(i\hbox {th}\) item
 (vi)
\(S_{i}\)= shortage level for the \(i\hbox {th}\) item
 (vii)
\(D_{i}(t)\)=demand rate of \(i\hbox {th}\) item, which is a function of inventory level at time t
 (viii)
\(\vartheta\)=inventory level elasticity of demand rate(\(0 \le \vartheta <1\))
 (ix)
\(t_{1i}\)= the time at which the inventory level reach zero for \(i\hbox {th}\) item(a decision variable)
 (x)
\(t_{2i}\)= the length of period during which are allowed for \(i\hbox {th}\) item
 (xi)
\(T_{i}\)= the length of the inventory cycle, hence \(T_{i}=t_{1i}+t_{2i}\)(a decision variable)
 (xii)
\(H_{i}[q_{i}(t)]\)= holding cost for the \(i\hbox {th}\) item, which is function of inventory level at time t
 (xiii)
\(\gamma\)=holding cost elasticity(\(\gamma \ge 1\))
 (xiv)
\(h_{i}\)= scaling constant for holding cost
 (xv)
\(w_{i}\) = storage space per unit quantity for the \(i\hbox {th}\) item
 (xvi)
B = budget available for replenishment
 (xvii)
F = available storage space in the inventory system
4.2 Assumption
 (i)
The replenishment rate is infinite and the leadtime zero.
 (ii)
The time horizon of the inventory system is infinite.
 (iii)
Shortage are allowed and during the stockout 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.
 (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:where \(\lambda _{i}> 0\) and \(0 \le \vartheta < 1\).$$\begin{aligned} D_{i}(t)=D_{i}[q_{i}(t)]= \left\{ \begin{array}{ll} \lambda _{i} [q_{i}(t)]^{\vartheta }, & \quad \text {if } 0 \le t \le t_{1i},\, q_{i}(t) \ge 0; \\ \lambda _{i}, & \quad \text {if } \, t_{1i} < t \le T_{i},\,q_{i}(t)> 0; \end{array} \right. \end{aligned}$$
 (v)
the holding cost is nonlinear 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
The ordering cost per cycle for \(i\hbox {th}\) item is \(A_{i}\)
5.1 Fuzzyrough inventory model
Therefore, the proposed fuzzy rough multiobjective problem shown in Eq. (28), which is the fuzzy representation form of the proposed crisp model Eq. (27).
5.2 Solution methodology
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 highend, midend, and lowend 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 (highend), II (midend), III (lowend), 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).
Input fuzzy rough cost parameters
Item  I  II  III 

\(\widehat{\tilde{s}}_{i}\)  \((\widehat{s}_{1}8.0, \widehat{s}_{1}, \widehat{s}_{1}+10)\)  \((\widehat{s}_{2}7.0, \widehat{s}_{2}, \widehat{s}_{2}+10)\)  \((\widehat{s}_{3}5.0, \widehat{s}_{3}, \widehat{s}_{3}+8.0)\) 
\(\widehat{s}_{1} \in ([32, 38] [31, 40])\)  \(\widehat{s}_{2} \in ([36, 40] [34, 42])\)  \(\widehat{s}_{3} \in ([35, 36] [31, 40])\)  
\(\widehat{\tilde{h}}_{i}\)  \((\widehat{h}_{1}2.0, \widehat{h}_{1}, \widehat{h}_{1}+2.0)\)  \((\widehat{h}_{2}1.5, \widehat{h}_{2}, \widehat{h}_{2}+3.0)\)  \((\widehat{h}_{3}2.0, \widehat{h}_{3}, \widehat{h}_{3}+1.0)\) 
\(\widehat{h}_{1} \in ([6.0, 7.5] [5.0, 8.0])\)  \(\widehat{h}_{2} \in ([7.0, 8.0] [6.0, 9.0])\)  \(\widehat{h}_{3} \in ([8.0, 9.5] [7.0, 10.0])\)  
\(\widehat{\tilde{c}}_{i}\)  \((\widehat{c}_{1}1.0, \widehat{c}_{1},\widehat{c}_{1}+2.0)\)  \((\widehat{c}_{2}1.5, \widehat{c}_{2}, \widehat{c}_{2}+1.0)\)  \((\widehat{c}_{3}2.0, \widehat{c}_{3}, \widehat{c}_{3}+3.0)\) 
\(\widehat{c}_{1} \in ([12, 14] [11, 15])\)  \(\widehat{c}_{2} \in ([16, 18] [15, 19])\)  \(\widehat{c}_{3} \in ([12, 14] [10, 15])\)  
\(\widehat{\tilde{c}}_{1i}\)  \((\widehat{c}_{11}3.0, \widehat{c}_{11}, \widehat{c}_{11}+3.5)\)  \((\widehat{c}_{12}2.0, \widehat{c}_{12}, \widehat{c}_{12}+4.0)\)  \((\widehat{c}_{13}1.0, \widehat{c}_{13}, \widehat{c}_{13}+3.0)\) 
\(\widehat{c}_{11} \in ([10, 12] [9.0, 16])\)  \(\widehat{c}_{12} \in ([13, 14] [12, 18])\)  \(\widehat{c}_{13} \in ([9.0, 11] [8.0, 17])\)  
\(\widehat{\tilde{c}}_{3i}\)  \((\widehat{c}_{31}10, \widehat{c}_{31}, \widehat{c}_{31}+13)\)  \((\widehat{c}_{32}12, \widehat{c}_{32}, \widehat{c}_{32}+14)\)  \((\widehat{c}_{33}8.0, \widehat{c}_{33}, \widehat{c}_{33}+12)\) 
\(\widehat{c}_{31} \in ([40, 42] [38, 52])\)  \(\widehat{c}_{32} \in ([43, 45] [40, 58])\)  \(\widehat{c}_{33} \in ([35, 37] [34, 46])\)  
\(\widehat{\tilde{w}}_{i}\)  \((\widehat{w}_{1}1.0, \widehat{w}_{1}, \widehat{w}_{1}+1.0)\)  \((\widehat{w}_{2}1.0, \widehat{w}_{2}, \widehat{w}_{2}+1.5)\)  \((\widehat{w}_{3}1.5, \widehat{w}_{3}, \widehat{w}_{3}+2.0)\) 
\(\widehat{w}_{1} \in ([3.0, 3.5] [2.0, 4.0])\)  \(\widehat{w}_{2} \in ([4.0, 4.5] [3.0, 5.0])\)  \(\widehat{w}_{3} \in ([4.0, 5.0] [3.5, 6.0])\)  
\(\widehat{\tilde{A}}_{i}\)  \((\widehat{A}_{1}8.0, \widehat{s}_{1}, \widehat{A}_{1}+8.0)\)  \((\widehat{A}_{2}5.0, \widehat{A}_{2}, \widehat{A}_{2}+7.0)\)  \((\widehat{A}_{3}8.0, \widehat{A}_{3}, \widehat{A}_{3}+12)\) 
\(\widehat{A}_{1} \in ([40, 80] [30, 90])\)  \(\widehat{A}_{2} \in ([45, 85] [25, 95])\)  \(\widehat{A}_{3} \in ([50, 90] [25, 100])\) 
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 \(\mu _{1}, \mu _{2}\).
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 
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 
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
 (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).
 (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).
 (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).
 (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).
 (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).
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 multiobjective multiitem inventory model under both stockdependent 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. TrPos 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 multiobjective 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, timedependent holding cost and others.
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