Economic ordering quantity inventory model with verhulst’s demand under fuzzy uncertainty for geographical market

Abstract

Inventory plays an important role in the production process. One of the primary reasons why inventory management modeling is essential for the industry is because it will suffer immensely if there are insufficient food products to stock during the shutdown period. By determining the combined optimal cost of the retailers and wholesalers, this research significantly improves the service of the supply chain from wholesaler to retailer. The stochastic number for the imperfect perishable items is provided in this inventory study. By altering the parameter values, the uniform distribution is used to calculate these damaged items. This approach identifies the backordering quantity for both regular and uncertain fish band circumstances. The cost of maintaining the inventory will rise significantly of increased wastage due to a rise in deteriorating, which will result in the loss of perishable food items. The primary goal of this research paper is to transport them without being destroyed until they reach their desired consumers. By determining the back ordering quantity during a shutdown, one can decrease the overall expenses incurred by the retailers. These computational complexity measures are proven in a fuzzy uncertain environment. The main goal of this paper is to analyze the variation of demand during the unanticipated period and find the optimum total cost of the perishable products. The growth of production in a particular area at a particular time, interconnect with another large number of products in the same area and is calculated by Verhulst’s demand with time depended on proficiency rate. Concerning the existing Verhulst’s demand pattern for the production process, this paper introduced that for perishable items in a fuzzy unanticipated situation. A bountiful system analysis is performed to find the cost function under fuzzy environment and the sensitivity analysis is carried out to perform the key representation constant.

Keywords

Perishable items unanticipated period verhulst’s demand hexagonal fuzzy number backorders geographical market fish ban period

1 Introduction

The main phenomenon of inventory management is stock level. The rate of consumption of deteriorating food products depends on time. In real life, fish is the leading food for the people in the coastal area. This proposed method extends the application of the production process in the fishing industry. Cite selection area for numerical validations in the east coast fishing area is illustrated in the proposed method. The fishing industry plays a very important role in the seashore area. Aquacultures practices vary depending on culture, environment, society, and the sources of fish. Due to the deterioration nature of fish, the demand rate may go up and down during the fish ban period. To avoid overfishing and for fish breeding, Tamil Nadu Marine Fisheries Regulation Act (TNMFRA) allotted 45 days to ban fishing at different coastal areas in different districts of Tamil Nadu.

During the 45 days lockdown period, small trawlers are going fishing in the first two or three yachting miles offshore. The demand for these fish products are very high during these 45 days fish ban period, because all the remaining big fiber boats, gill net boats are not allowed to yachting as is shown in Fig. 1.

Fig. 1

Small-scale and Large-scale trawlers.

These small-scale trawlers leave by 2 AM and they return on the same day. They got fish varieties of different types by the gill net boats (big trawlers) entirely different from deep-sea varieties. These fish species are fresh and the demand is very high and there is no need to preserve them as is shown in Fig. 2. The demand rate of these items rapidly increases during the 45 days and beyond the certain time, the rate of demand is reduced to zero. Then it will start again during the next year fish ban period.

Fig. 2

Demand after Fish ban period.

By using the previous year’s data, the demand can be assumed. But, by applying the mathematical Verhulst’s demand pattern for the real-time analysis, the exact demand rate for the separate species in the surrounding environment can be predicted. From that the total cost of the entire process can be reduced. Also, the inventory level, as well as the back-ordered quantity for the fish products can be found during the lockdown situation.

2 Literature survey

Moncer Hariga [11] explained the linear and exponential demand pattern along with the deterioration rate and completely back-ordered with three replenishment policies. S.K. Pattanayak [16] dealt with the fisheries resources and diversification of a well-planned modern industry of fishing of Karnataka. He assessed the reason for the fluctuations of total cost in fisheries.

Dennis Fok et al. [5] dealt with weekly seasonality that decomposed the goods in summer and winter. The non-linear price effects are explained by Markov - chain. Sunil Sabat et al. [17] analyzed urban, rural, and semi-urban areas in the production of fish consumption. Also, they analyzed the demand for fish products in all three areas with socio-economic variables. Mingbao cheng et al. [12] described the trapezoidal demand rate in production inventory management for deteriorated items. Chun-Tao Chang et al. [4] considered the trapezoidal type of demand rate for perishable inventory. Himani Dem et al. [7] flourished the imperfect production process for amelioration inventory system. They conclude that the result of their research was more realistic while selling the imperfect items with a discount price which results in a maximized profit. Chakraborty et al. [3] explained the concept of inventory production in a fuzzy environment and assumed the decision variables as a hexagonal fuzzy number. They explained different defuzzification methods which is a motivation to assume the hexagonal fuzzy number as a decision variable and to compare it with the crisp case. Taleizadeh and Dehkordi [2] discussed the same inventory management for the deteriorated items and applied the normal distribution for the non-defective items. The extension of this paper in a fuzzy environment gives the optimal solution in total cost.

Chauhan and Singh [1] derived demand for the inventory model and compared this with ramp type demand. They explained the whole concept of the life span of the deteriorating products. Parvathi and Gajalakshmi [15] explained the concept of Verhulst’s demand using hexagonal fuzzy numbers. They separated back-ordered cost as linear and fixed and conclude that the inventory cost is minimum. Aswathy et al. [13] analyzed the revenue functions for fish manufacturing in fisheries.

Ganesan et al. [6] dealt with the stock trend prediction in fuzzy cognitive maps. This paper deals with decision-making in stock trends using the previous predictions. K. Jeganathan et al. [10] considered the model with stock depended demand and conclude the optimal total cost of the queuing inventory. Broumi et al. [18] compared the various algorithms for uncertain environment and finally conclude the best algorithm to solve uncertain environment. Yusuf Ibrahim Gwanda et al. [21] developed the Verhulst’s demand with time depended deterioration rate and demonstrated with three optimization steps to attain the maximum profit in the EOQ model.

Kuppulakshmi et al. [19] derived the total cost of the production of perishable fish items. K. Jeganathan et al. [9] derived a paper in queue dependent service rate (QDSR) and retrial facility is followed. In this paper, rework is considered for the deteriorated product.

Jeganathan et al. [8] derived the model in queue depended on service for the inventory with a stochastic process. Kuppulakshmi et al. [20] derived the imperfect production process for the perishable items for the constant demand. The extension of this paper in Verhulst’s demand is discussed in a fuzzy environment and got the optimal solution in the present study. Nagar H et al. [14] evaluated the total inventory cost of the production plan in a fuzzy environment. Under the situations of (i) fully backordered (shutdown) and (ii) partially backordered (Regular days), this research was designed to determine the best ordered size and appropriate inventory level (normal geographical market). By the faulty measurement in the demand, some sellers are unable to fulfill the buyers in both situations. Food is considered to be a vital item, and demand for it will rise, particularly during a shutdown. Finding the best method for storage food that can be supplied when needed is important in such a critical situation. One instance where all the nations learned the value of storing food for the future is during a lockdown. One of the primary reasons why inventory management planning is required for the industry is that, if there are insufficient food products to stock or if the stocks are not used for a longer period of time, the industry will suffer significant loss. In this study, the optimum total cost during lockdown and under normal conditions could be determined by calculating the appropriate backordered quantity and inventory level for fish product using Verhulst’s demand.

While modulating the production during the unanticipated period, the demand for perishable products sores exponentially. Verhulst’s demand pattern is used to find the growth and maturity stage of perishable products, which ensures the unhindered supply of perishable products during the unanticipated period. When parameters are changed, the carrying capacity of perishable products and ordering cost, Inventory holding cost, and back-ordering cost variations are explained and these variations are shown for uncertain measures.

In most papers, cost functions are treated as fuzzy numbers. But in this paper, during unanticipated situations, the perishable rate is treated as the hexagonal fuzzy number for the production process. In order to characterize the behavior of deteriorating food products, this paper is developed for profit-maximizing inventory model that is motivated by the Verhulst demand rate model. The wholesale management deal with those perishable items could use this model to calculate the total cost and cycle time under the proposed Verhulst’s demand model. This analysis plays a vital role to demonstrate the effectiveness of this model.

3 Basic definitions

Basic definitions are launched by Zadeh in 1965.

3.1 Definition [20]

A fuzzy set [u_∝, v_∝] where 0 ≤ ∝ ≤ 1 and a < b determined on R is termed as the fuzzy interval when its membership function is given by $μ_{[a_{\propto}, b_{\propto}]} = {0, \begin{matrix} \propto, u ⩽ x ⩽ v \\ or else \end{matrix}}$

3.2 Definition [20]

A hexagonal fuzzy number $\tilde{Δ}$ =(x, y, z, s, t, u) is represented with a membership function $μ_{\overset{⌣}{A}}$ . $μ_{\overset{⌣}{Δ}} = {\begin{matrix} L_{1} (p) = \frac{1}{2} (\frac{p - x}{y - x}), & x ⩽ p ⩽ y \\ L_{2} (p) = \frac{1}{2} + \frac{1}{2} (\frac{p - y}{z - s}), & y ⩽ p ⩽ z \\ 1, & z ⩽ p ⩽ s \\ R_{1} (p) = 1 - \frac{1}{2} (\frac{p - s}{t - s}), & s ⩽ p ⩽ t \\ R_{2} (p) = \frac{1}{2} (\frac{u - p}{u - t}), & t ⩽ p ⩽ u \\ 0, & otherwise \end{matrix}}$

The ∝- cut on $\tilde{Δ}$ = (a, b, c, d, e, f), 0 ≤ ∝ ≤ 1 is Δ(∝) = [Δ_L(∝) , Δ_R(∝)]. where $Δ_{L_{1}} (\propto) = x + (y - x) \propto = {L_{1}}^{- 1} (\propto)$ $Δ_{L_{2}} (\propto) = y + (z - y) \propto = {L_{2}}^{- 1} (\propto)$ $Δ_{R_{1}} (\propto) = s + (t - s) \propto = {R_{1}}^{- 1} (\propto)$ $Δ_{R_{2}} (\propto) = u (u - t) \propto = {R_{2}}^{- 1} (\propto)$ Hence, $L^{- 1} (\propto) = \frac{{L_{1}}^{- 1} (\propto) + {L_{2}}^{- 1} (\propto)}{2} = \frac{x + y + (z - x) \propto}{2}$ $R^{- 1} (\propto) = \frac{{R_{1}}^{- 1} (\propto) + {R_{2}}^{- 1} (\propto)}{2} = \frac{t + u + (s - u) \propto}{2}$

3.3 Definition [20]

Graded Mean Representation Method (GMRM) is used for defuzzification. If $\tilde{Δ} = (x, y, z, s, t, u)$ is a hexagonal fuzzy number, then $P (\tilde{Δ}) = \frac{1}{12} [x + y + 2 z + 2 s + 3 t + u]$

Hexagonal fuzzy number provides valid information and easy understanding of vague information. Hexagonal fuzzy numbers provide valid information and an easy understanding of indefinite values. Defuzzification of hexagonal fuzzy gives an accurate value compared to other triangular and trapezoidal fuzzy numbers. The graded mean integration method is a simple and easy method for defuzzification.

4 Assumption

For the fish ban period, Verhulst’s demand is considered

After the fish ban period, the demand for the fish product reduces automatically.

Backlogging is allowed

Shortage is divided into two types (i) during fish ban period and (ii) normal period

In a fuzzy environment, deterioration cost, the mean of defective items, and the variance of defective items are considered as hexagonal fuzzy numbers.

5 Notations

Table 1
Notation of the model

D_F Constant Verhulst demand

D _F(0) Initial stage population of perishable products at beginning of fish ban period (tones)

F _r The maximum growth rate of fish products during the fish ban period

T Fish ban period (days)

E _K Transferring magnitude of the surrounding environment (tones)

φ $\frac{F_{r}}{E_{K}}$ (Constant of proportion)

I Optimal ordered quantity of perishable product.

B Back-ordered Quantity per year

F _c Ordering cost for a single item.

D _c The cost of decaying for a single item.

H _c Stock holding/year

Q Ordered quantity of perishable products.

γ Good quality items (r≤γ≤1)

f (γ) PDF of perishable items (m-mean, v- variance)

S _n Back ordering cost (without fish ban period)

S _L Back ordering cost (fish ban period)

θ Perishable rate

θ _c Perishable cost

O _Q γI

T $\frac{O_{Q}}{D_{F}} (Total cycle time)$

Ψ _L ρα_L +(1 - ρ)(μ_L + F_s - F_p)

Ψ ρα +(1 - ρ) μ

F_s Retail price of one item.

F _p Procure cost for one unit of item.

μ Cost of penalty (normal run time)

E(.) Expected value

TC(I, B) Optimal total cost

α_L Perishable item back ordering cost (fish ban)

α Perishable item back ordering cost (normal run time)

μ_L Penalty cost of items lost (fish ban) ρ Proportion of backordered parameter

D_F	Constant Verhulst demand
D _F(0)	Initial stage population of perishable products at beginning of fish ban period (tones)
F _r	The maximum growth rate of fish products during the fish ban period
T	Fish ban period (days)
E _K	Transferring magnitude of the surrounding environment (tones)
φ	$\frac{F_{r}}{E_{K}}$ (Constant of proportion)
I	Optimal ordered quantity of perishable product.
B	Back-ordered Quantity per year
F _c	Ordering cost for a single item.
D _c	The cost of decaying for a single item.
H _c	Stock holding/year
Q	Ordered quantity of perishable products.
γ	Good quality items (r≤γ≤1)
f (γ)	PDF of perishable items (m-mean, v- variance)
S _n	Back ordering cost (without fish ban period)
S _L	Back ordering cost (fish ban period)
θ	Perishable rate
θ _c	Perishable cost
O _Q	γI
T	$\frac{O_{Q}}{D_{F}} (Total cycle time)$
Ψ _L	ρα_L +(1 - ρ)(μ_L + F_s - F_p)
Ψ	ρα +(1 - ρ) μ
F_s	Retail price of one item.
F _p	Procure cost for one unit of item.
μ	Cost of penalty (normal run time)
E(.)	Expected value
TC(I, B)	Optimal total cost
α_L	Perishable item back ordering cost (fish ban)
α	Perishable item back ordering cost (normal run time)
μ_L	Penalty cost of items lost (fish ban) ρ Proportion of backordered parameter

6 Mathematical modeling

During the fish ban periods, the demand for fish products is very high. Verhulst’s demand is considered in this model, so we can find the demand rate very easily. The total ban period is assumed to be 45 days. At time t = 0, the total perishable products are collected from the vendor and assumed as Q. Demand is satisfied by the customer by the wholesale or retail process (i.e.) when demand is increasing due to uncertain situation. The proportion parameter (1-ρ) for the uncertain situation and the proportion parameter ρ for the normal situation is explained in Fig. 3. The Verhulst’s demand is given as, $\frac{d D_{F}}{dt} = \frac{F_{r}}{E_{K}} D_{F} {E_{K} - D_{F}} and \frac{F_{r}}{E_{K}} = φ = 0.01$ (1) $D_{F} = \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}}$ (2) When t = 0 $D_{F} (0) = \frac{C E_{K}}{1 + C}$ (3) From the above equation, we get the constant $C = \frac{D_{F} (0)}{E_{K} - D_{F} (0)}$ (4)

Fig. 3

Fish ban period inventory control system using Verhulst’s demand (D_F).

$\begin{matrix} Holdind cost = \\ {\frac{(O_{Q} - B)}{2 D_{F}}}^{2} H_{c} = {\frac{(O_{Q} - B)}{2 C e^{φ t} E_{K}}}^{2} (1 + C e^{φ t}) H_{c} \end{matrix}$ (5) $Shortage cost = Ψ_{L} B + Ψ \frac{B^{2}}{2 C e^{φ t} E_{K}} (1 + C e^{φ t})$ (6)

Ψ_L is a back ordering cost independent of fish ban period (α_L) and μ_L + F_s - F_p. Shortage cost is also including the revenue lost and back ordering cost for normal period (Ψ). The total production run time is defined by the equation, $T = \frac{O_{Q}}{C e^{φ t} E_{K}} (1 + C e^{φ t}) = \frac{γ I (1 + C e^{φ t})}{C e^{φ t} E_{K}}$ (7) The optimal cycle time per year is given as $E (T) = E (\frac{O_{Q (1 + C e^{φ t})}}{C e^{φ t} E_{K}}) = \frac{mI (1 + C e^{φ t})}{C e^{φ t} E_{K}}$ (8) The perishable cost for the decaying items depending on the perishable rate and with the inventory is given as $D_{c} = θ_{c} I and θ_{c} = θ C_{d}$ (9) The total cost for the period includes the normal period and the fish ban period and is formulated as,

$\begin{matrix} TC (O_{Q}, B) = \\ F_{c} + {\frac{(O_{Q} - B)}{2}}^{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} \\ H_{c} + Ψ_{L} B + Ψ \frac{B^{2}}{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} + θ_{c} I \end{matrix}$ (10) The expected total cost is given as,

$\begin{matrix} E (TC) = \int_{r}^{1} [F_{c} + {\frac{(γ I - B)}{2}}^{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} H_{c} \\ + Ψ_{L} B + Ψ \frac{B^{2}}{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} + θ_{c} I] f (γ) d γ \end{matrix}$ (11)

$\begin{matrix} = F_{c} + \int_{r}^{1} [{\frac{(γ I - B)}{2}}^{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} H_{c}] f (γ) d γ \\ + Ψ_{L} B + Ψ \frac{B^{2}}{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} + θ_{c} I \end{matrix}$ (12) here $\int_{r}^{1} γ^{2} f (γ) d γ = m^{2} + v and$ (13) $\int_{r}^{1} γ f (γ) d γ = m$ (14)

It follows normal distribution with mean m and variance m² + v

$\begin{matrix} E (TC) = F_{c} + \frac{(m^{2} + v) I^{2} H_{c}}{2} \frac{(1 + C e^{φ t})}{C e^{φ t} E_{K}} \\ + \frac{mIB H_{c} (1 + C e^{φ t})}{C e^{φ t} E_{K}} + Ψ_{L} B + \frac{Ψ B^{2}}{2 D_{F}} + \frac{B^{2} H_{c}}{2 D_{F}} \\ + mI D_{F} - {(\frac{C e^{φ t} E_{K}}{1 + C e^{φ t}})}^{2} \end{matrix}$ (15)

The expected total cost over one year is given as, $Expected total cost = \frac{E (TC)}{E (T)} E (T) = \frac{mIC e^{φ t} E_{K}}{1 + C e^{φ t}}$ (16)

Therefore, the cost function is

$\begin{matrix} TC (I, B) = \frac{F_{c}}{mI} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} + \frac{(m^{2} + v) I}{2 m} H_{c} \\ - B H_{c} + \frac{Ψ_{L} BC e^{φ t} E_{K}}{mI (1 + C e^{φ t})} \\ + \frac{(Ψ + H_{c}) B^{2}}{2 mI} + \frac{θ_{c} C e^{φ t} E_{K}}{m (1 + C e^{φ t})} \end{matrix}$ (17)

The optimum fish ordered quantity and the optimum back ordered quantity for both the normal situation and the fish ban period together is given as,

To find the optimum solution for I^*

$\begin{matrix} \frac{\partial TC (I, B)}{\partial I} = - \frac{F_{c}}{m I^{2}} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} + \frac{(m^{2} + v)}{2 m} H_{c} \\ - \frac{Ψ_{L} B}{m I^{2}} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} - \frac{(Ψ + H_{c}) B^{2}}{2 m I^{2}} \end{matrix}$ (19) $I^{*} = \sqrt{\frac{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} + (Ψ + H_{c}) B^{2} + 2 Ψ_{0} B \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}}}{(m^{2} + v) H_{c}}}$ (20) By substituting the I^* in the Equation (11) $TC (I, B) = (\sqrt{\frac{(m^{2} + v)}{m^{2}}})$ $\begin{matrix} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{2} H_{c} + 2 B \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L} H_{c}} \\ + \frac{θ_{c} D_{F}}{m} - B H_{c} \end{matrix}$ (21)

6.1 criteria 1

$\begin{matrix} \frac{\partial TC (I, B)}{\partial B} = \sqrt{\frac{(m^{2} + v)}{m^{2}}} \frac{1}{2} \\ \frac{2 B (Ψ + H_{c}) H_{c} + 2 Ψ_{L} D_{F} H_{c}}{\sqrt{2 F_{c} D_{F} H_{c} + (Ψ + H_{c}) B^{2} H_{c} + 2 B Ψ_{L} D_{F} H_{c}}} - H_{c} \end{matrix}$ (22) $= H_{c} \sqrt{\frac{(m^{2} + v)}{m^{2}}} \frac{B (Ψ + H_{c}) + Ψ_{L} D_{F}}{\sqrt{2 F_{c} D_{F} H_{c} + (Ψ + H_{c}) B^{2} H_{c} + 2 B Ψ_{L} D_{F} H_{c}}} - H_{c}$ (23) $\begin{matrix} = H_{c} [\sqrt{\frac{(m^{2} + v)}{m^{2}}} \\ \frac{B (Ψ + H_{c}) + Ψ_{L} D_{F}}{\sqrt{2 F_{c} D_{F} H_{c} + (Ψ + H_{c}) B^{2} H_{c} + 2 B Ψ_{L} D_{F} H_{c}}} - 1] \end{matrix}$ (24) $= \frac{\begin{matrix} \frac{(m^{2} + v)}{m^{2}} (B^{2} {(Ψ + H_{c})}^{2} + {Ψ_{L}}^{2} {D_{F}}^{2} + 2 B (Ψ + H_{c}) Ψ_{L} D_{F}) - \\ 2 F_{c} D_{F} H_{c} - B^{2} (Ψ + H_{c}) H_{c} - 2 B Ψ_{L} D_{F} H_{c} \end{matrix}}{Δ (B) (\sqrt{\frac{(m^{2} + v)}{m^{2}}} B (Ψ + H_{c}) + \sqrt{\frac{(m^{2} + v)}{m^{2}}} Ψ_{L} F_{c} + Δ (B))} = 0$ (25) $= \frac{\prod (B)}{Δ (B) (\sqrt{\frac{(m^{2} + v)}{m^{2}}} B (Ψ + H_{c}) + \sqrt{\frac{(m^{2} + v)}{m^{2}}} Ψ_{L} F_{c} + Δ (B))} = 0$ (26) Were $Δ (B) = \sqrt{2 F_{c} D_{F} H_{c} + (Ψ + H_{c}) B^{2} H_{c} + 2 B Ψ_{L} D_{F} H_{c}}$ (27) $\begin{matrix} \prod (B) = \\ \begin{matrix} \frac{(m^{2} + v)}{m^{2}} (B^{2} {(Ψ + H_{c})}^{2} + {Ψ_{L}}^{2} {D_{F}}^{2} + 2 B (Ψ + H_{c}) Ψ_{L} D_{F}) - \\ 2 F_{c} D_{F} H_{c} - B^{2} (Ψ + H_{c}) H_{c} - 2 B Ψ_{L} D_{F} H_{c} \end{matrix} \end{matrix}$ (28)

To find the optimum solution for B^* (criteria 1) $\begin{matrix} \frac{\partial TC (I, B)}{\partial B} = \\ \frac{\prod (B)}{Δ (B) ([\sqrt{\frac{(m^{2} + v)}{m^{2}} (Ψ + H_{c}) B + \frac{(m^{2} + v)}{m^{2}} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L}}] + Δ (B))} \\ = 0 \end{matrix}$ (29) $\begin{matrix} \prod (B) = [\frac{(m^{2} + v)}{m^{2}} ({(Ψ + H_{c})}^{2}) - (Ψ + H_{c}) H_{c}] B^{2} \\ + [\frac{(m^{2} + v)}{m^{2}} 2 Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} (Ψ + H_{c}) - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L} H_{c}] B \\ + [\frac{(m^{2} + v)}{m^{2}} {Ψ_{L}}^{2} {\frac{C e^{φ t} E_{K}}{1 + C e^{φ t}}}^{2} - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} F_{c}] \end{matrix}$ (30) $a = [\frac{(m^{2} + v)}{m^{2}} ({(Ψ + H_{c})}^{2}) - (Ψ + H_{c}) H_{c}]$ (31) $b = [\frac{(m^{2} + v)}{m^{2}} 2 Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} (Ψ + H_{c}) - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L} H_{c}]$ (32) $c = [\frac{(m^{2} + v)}{m^{2}} {Ψ_{L}}^{2} {(\frac{C e^{φ t} E_{K}}{1 + C e^{φ t}})}^{2} - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} F_{c}]$ (33) $\begin{matrix} Δ = b^{2} - 4 ac = \\ {[\frac{(m^{2} + v)}{m^{2}} 2 Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} (Ψ + H_{c}) - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L} H_{c}]}^{2} \\ - 4 [\frac{(m^{2} + v)}{m^{2}} ({(Ψ + H_{c})}^{2}) - (Ψ + H_{c}) H_{c}] \\ [\frac{(m^{2} + v)}{m^{2}} {Ψ_{L}}^{2} {(\frac{C e^{φ t} E_{K}}{1 + C e^{φ t}})}^{2} - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} F_{c}] \end{matrix}$ (34)

The formula $\frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}$ is used to find the root of ∏(B). $B^{*} = \frac{- \frac{(m^{2} + v)}{m^{2}} 2 Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} (Ψ + H_{c}) + 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L} H_{c} + \sqrt{Δ}}{2 \frac{(m^{2} + v)}{m^{2}} ({(Ψ + H_{c})}^{2}) - (Ψ + H_{c}) H_{c}}$ (35) Or $B^{*} = \frac{- \frac{(m^{2} + v)}{m^{2}} 2 Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} (Ψ + H_{c}) - 2 \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} Ψ_{L} H_{c} + \sqrt{Δ}}{2 \frac{(m^{2} + v)}{m^{2}} ({(Ψ + H_{c})}^{2}) - (Ψ + H_{c}) H_{c}}$ (36) Considering the positive quantity for B^* (equation 33). By using MATLAB software, the optimum value TC(I, B) is calculated as follows $\begin{matrix} TC (I, B) = \sqrt{\frac{(m^{2} + v)}{m^{2}}} \\ \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c} \end{matrix}$ (37)

6.2 Criteria (2)

To prove the criteria (2) $\frac{\partial TC (I, B)}{\partial I} = - \frac{F_{c} D_{F}}{m I^{2}} + \frac{(m^{2} + v)}{2 m} H_{c} - \frac{Ψ_{L} B D_{F}}{m I^{2}} - \frac{(Ψ + H_{c}) B^{2}}{2 m I^{2}}$ (38) $\frac{\partial^{2} TC (I, B)}{\partial I^{2}} = 2 \frac{F_{c} D_{F}}{m I^{3}} + \frac{2 Ψ_{L} B D_{F}}{m I^{3}} + \frac{2 (Ψ + H_{c}) B^{2}}{m I^{3}} > 0$ (39) To prove criteria 2 $\frac{\partial^{2} TC}{\partial B \partial I} = - 2 B (Ψ + H_{c}) - \frac{2 Ψ_{L} D_{F}}{m I^{2}}$ (40) ${(\frac{\partial^{2} TC}{\partial B \partial I})}^{2} = \frac{4 m^{2} B^{2} I^{4} {(Ψ + H_{c})}^{2} + 4 {Ψ_{L}}^{2} {D_{F}}^{2} + 8 (Ψ + H_{c}) Ψ Ψ_{L} m I^{2} B D_{F}}{m I^{2}}$ (41) $\frac{\partial^{2} TC}{\partial B^{2}} = \frac{(Ψ + H_{c})}{mI} + \frac{Ψ_{L} D_{F}}{mI}$ (42) $(\frac{\partial^{2} TC}{\partial I^{2}} \frac{\partial^{2} TC}{\partial B^{2}}) - {(\frac{\partial^{2} TC}{\partial B \partial I})}^{2} = \frac{\begin{matrix} (Ψ + H_{c} + Ψ_{L} D_{F}) (2 B^{2} (Ψ + H_{c}) + 2 Ψ_{L} B D_{F} + 2 D_{F} H_{c}) \\ - \\ 4 (B^{2} m^{2} I^{4} {(Ψ + H_{c})}^{2} + 4 {Ψ_{L}}^{2} {D_{F}}^{2} + 8 (Ψ + H_{c}) Ψ Ψ_{L} m I^{2} B D_{F} \end{matrix}}{m^{2} I^{2}}$ (43) Provided that, $\begin{matrix} (Ψ + H_{c} + Ψ_{L} D_{F}) (2 B^{2} (Ψ + H_{c}) + 2 Ψ_{L} B D_{F} + 2 D_{F} H_{c}) > \\ 4 (B^{2} m^{2} I^{4} {(Ψ + H_{c})}^{2} + 4 {Ψ_{L}}^{2} {D_{F}}^{2} + 8 (Ψ + H_{c}) Ψ Ψ_{L} m I^{2} B D_{F} \end{matrix}$ For convexity it should satisfy (above criteria 2) $(i) \frac{\partial^{2} TC (I, B)}{\partial I^{2}} > 0$ (44) $(ii) (\frac{\partial^{2} TC}{\partial I^{2}} \frac{\partial^{2} TC}{\partial B^{2}}) - (\frac{\partial^{2} TC}{\partial B \partial I}) > 0$ (45) The total optimal cost TC (I, B) is minimum

6.3 Fuzzy model

The crisp model was developed and all parameters are assumed to be fixed and found by certainty. But in real case, it is not possible to find the accurate value. Therefore, this sample has been taken in a fuzzy circumstance. In fuzzy model, we take deterioration cost, mean of the defective items and the variance of defective item. $\tilde{m}$ =fuzzy defective mean $\tilde{υ}$ =fuzzy defective variance, $\tilde{θ_{c}}$ =fuzzy deterioration cost, suppose $\tilde{m} = (m_{1}, m_{2}, m_{3}, m_{4}, m_{5}, m_{6})$ , $\tilde{υ} = (υ_{1}, υ_{2}, υ_{3}, υ_{4}, υ_{5}, υ_{6})$ and $\tilde{θ_{c}} = ({θ_{c}}_{1}, {θ_{c}}_{2}, {θ_{c}}_{3}, {θ_{c}}_{4}, {θ_{c}}_{5}, {θ_{c}}_{6})$ are non-negative hexagonal fuzzy numbers (HFN). The total production cost is given below:

$\tilde{TC} (I, B) = {\tilde{(TC}}_{1} (I, B), {\tilde{TC}}_{2} (I, B), {\tilde{TC}}_{3} (I, B), {\tilde{TC}}_{4} (I, B), {\tilde{TC}}_{5} (I, B), {\tilde{TC}}_{6} (I, B))$ (46) where ${\tilde{TC}}_{1} (I, B) = \sqrt{\frac{({m_{1}}^{2} + v_{1})}{{m_{1}}^{2}}} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c_{1}} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c}$ (47) ${\tilde{TC}}_{2} (I, B) = \sqrt{\frac{({m_{2}}^{2} + v_{2})}{{m_{2}}^{2}}} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c_{2}} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c}$ (48) ${\tilde{TC}}_{3} (I, B) = \sqrt{\frac{({m_{3}}^{2} + v_{3})}{{m_{3}}^{2}}} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c_{3}} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c}$ (49) ${\tilde{TC}}_{4} (I, B) = \sqrt{\frac{({m_{4}}^{2} + v_{4})}{{m_{42}}^{2}}} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c_{4}} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c}$ (50) ${\tilde{TC}}_{5} (I, B) = \sqrt{\frac{({m_{5}}^{2} + v_{5})}{{m_{52}}^{2}}} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c_{54}} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c}$ (51) ${\tilde{TC}}_{6} (I, B) = \sqrt{\frac{({m_{6}}^{2} + v_{6})}{{m_{62}}^{2}}} \sqrt{2 F_{c} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c} + (Ψ + H_{c}) B^{* 2} H_{c} + 2 B^{*} Ψ_{L} \frac{C e^{φ t} E_{K}}{1 + C e^{φ t}} H_{c}} + θ_{c_{654}} \frac{C e^{φ t} E_{K}}{m (1 + C e^{φ t})} - B^{*} H_{c}$ (52) $\tilde{TC} (I, B) = \frac{1}{12} ({\tilde{TC}}_{1} (I, B) + 3 {\tilde{TC}}_{2} (I, B) + 2 {\tilde{TC}}_{3} (I, B) + 2 {\tilde{TC}}_{4} (I, B) + 3 {\tilde{TC}}_{5} (I, B) + {\tilde{TC}}_{6} (I, B))$ (53)

The defuzzification is done by GMRM is given as, $\tilde{B} = \frac{1}{12} (\tilde{B_{1}} + \tilde{3 B_{2}} + \tilde{2 B_{3}} + 2 \tilde{B_{4}} + \tilde{3 B_{5}} + \tilde{B_{6}})$ $\tilde{I} = \frac{1}{12} (\tilde{I_{1}} + \tilde{3 I_{2}} + \tilde{2 I_{3}} + 2 \tilde{I_{4}} + \tilde{3 I_{5}} + \tilde{I_{6}})$

7 Numerical result

7.1 Crisp model

The highly deteriorated nature of the perishable products affects the total cost of system. We consider m = 0.7, v = 0.01, H_c = $3/unit/year, α_L = 1.5, α=1, μ_L = $1.5/unit, μ=0/unit/year, μ_L + F_s - F_p = $2/unit, F_p = 200, F_c = 500, θ_c=0.3, t = 45days, E_K = 100 tones, F_r = 1, D_F(0) = 50 tones.

The solution of the crisp model is $TC (I, B) = $ 374.85, B^{*} = 49, I^{*} = 694$

7.2 Fuzzy mode

Suppose $\tilde{m}$ =(0.5 0.6 0.7 0.7 0.8 0.9), $\tilde{υ}$ =(0.0080 0.0090 0.01 0.01 0.0110 0.0120), $\tilde{θ_{c}}$ =(0.2980 0.2990 0.3 0.3 0.3010 0.3020) are hexagonal fuzzy numbers, then fuzzy backordered level, fuzzy inventory level and fuzzy total cost is given by, $\tilde{TC} {(I, B)}^{*} = $ 373, {\tilde{B}}^{*} = 52, {\tilde{I}}^{*} = 724$ .

8 Sensitivity analysis

While comparing the crisp model and the fuzzy environment, an increase in backorder level and inventory by level is not affected by the total cost. The optimal total cost of the manufacturing process is less while comparing the crisp case. The optimum result occurs in a fuzzy environment. From Table 2 the parameter change in backorder levels is explained in Fig. 5. In Fig. 5 carrying capacity (E_K) is increased for the parameters (–30% to +30%) so that the back ordering level also increases for increasing the carrying capacity. Ordering cost (F_c) is increased for the parameters (–30% to +30%) so that the back-ordering level is also increased for increasing the ordering cost. Holding Cost (H_c), back-ordered cost (α), fish ban period (t) is increased for the parameters (–30% to +30%) so the back- ordered level is also increased for increasing holding cost, back-ordered cost and fish ban period.

Table 2
Parameter changes in Backordered level

% –30 –20 –10 10 20 30

E _K 41.522 44.174 46.624 51.073 53.103 55.024

F _c 40.151 43.263 46.171 51.522 54 56.392

t 47.713 48.122 48.522 49.304 49.681 50.044

H _c 16 24 35 66 87.483 113

α 43.462 45.921 47.073 50.781 52.672 54.591

%	–30	–20	–10	10	20	30
E _K	41.522	44.174	46.624	51.073	53.103	55.024
F _c	40.151	43.263	46.171	51.522	54	56.392
t	47.713	48.122	48.522	49.304	49.681	50.044
H _c	16	24	35	66	87.483	113
α	43.462	45.921	47.073	50.781	52.672	54.591

Fig. 4

Verhulst’s demand during fish ban period.

Fig. 5

The Parameter changes in back ordered level.

From Table 3 the parameter change in inventory levels are explained as shown in Fig. 6. In Fig. 6, the carrying capacity (E_K) is increased for the parameters (–30% to +30%) so that the inventory level is also increased for increasing the carrying capacity. Ordering cost (F_c) is increased for the parameters (–30% to +30%) so that the inventory level is also increased for increasing the ordering cost. Holding cost (H_c) and back-ordered cost (α) is increased for the parameters (–30% to +30%), so the inventory level also increases for increasing holding cost and back-ordered cost. From Table 4, the parameter change in total cost are explained and is shown in Fig. 7. In Fig. 7, the carrying capacity (E_K) is increased for the parameters (–30% to +30%) so that the total cost increases for increasing the carrying capacity. Ordering cost (F_c) is increased for the parameters (–30% to +30%) so that the total cost also increases for increasing the ordering cost. Back ordered cost (α) is increased for the parameters (–30% to +30%) so the total cost also increases for increasing back ordered cost. Deterioration cost (θ_c) and fish ban period (t) is increased for the parameters (–30% to +30%) so the total cost also increases for an increase in the deterioration cost.

Fig. 6

Parameter change in inventory level.

Fig. 7

Parameter change in total cost.

Table 3

Parameter change in Inventory level (I)

%	–30	–20	–10	10	20	30
E _K	574.530	616.424	655.990	729.521	763.984	797.162
F _c	586.711	624.583	660.070	725.432	755.823	784.943
t	673.782	680.492	687.101	699.973	706.221	712.364
H _c	524.563	572.951	627.591	778.034	890.072	1041
α	681.124	685.164	689.314	697.990	702.524	707.117

Table 4

Parameter change in Total cost (TC)

%	–30	–20	–10	10	20	30
E _K	304.142	328.742	352.254	396.670	417.812	438.351
F _c	324.794	342.524	359.144	389.772	404.023	417.682
t	362.912	366.952	370.933	378.703	382.494	386.223
θ _c	366.994	369.614	372.233	377.463	380.082	382.694
α	382.290	379.800	377.321	372.392	369.953	367.521

Table 5

Comparative study

Existing research	Parameter	Development and consequence
(V Kuppulakshmi and C Sugapriya 2020)	Perishable fish items are produced during the yachting	Perishable fish items are produced and parameters are treated as hexagonal fuzzy numbers (HFN).
(V. Kuppulakshmi and C Sugapriya, Deivanayagam Pillai 2021)	Reworkable items, back ordering and perishable items.	The extension of the paper with Verhulst’s demand of the fishes discussed in the present paper. Better results occurred in this paper during lockdown and vessel breakdown in fuzzy environment.
(Nagar and Surana 2015)	Pentagonal fuzzy number, reworkable items, back ordering and Deteriorating item	The same reworkable rate is treated as hexagonal fuzzy number and gives the better result in fuzzy environment.
(Taleizadeh AA and Dehkordi NZ 2017)	Partial backorders, imperfect production.	Partial back ordering and full backordering are considered in the present paper for perishable items. The optimal solution for the total cost is obtained in the present paper while applying hexagonal fuzzy number.

9 Conclusion

The interpretation of this analysis is shown in both the fish ban period with the uncertain lockdown situation and the normal situation. The carrying capacity of the surrounding environment plays a vital role in Verhulst’s demand. In this paper, the total cost of the fuzzy environment is more feasible when compared with the crisp model. Even though the back-ordered quantity and the inventory levels are high, the expected cost of the production cycle is minimized while comparing the result with fuzzy. The risk of population broadening and the ravenous use of natural resources are affecting the struggle among the manufacture of fish market, policymaker of the retailers during the fish ban period, and wholesale dealers.

By using Verhulst’s demand pattern, the effect of increasing and decreasing demand of fish in an uncertain situation and the normal period are determined. This pattern is used to calculate the total cost spent by the merchants in the coastal area. These data are collected and the changes are resolute, which are used to compare the cost spent by both retailers and wholesale merchants of the coastal area. So, they can analyze the economic growth by using this model. It is useful to make decisions in the manufacturing process. Fuzzy parameters give a more accurate value than the crisp case. Because of the prior thinking of demand function, how much they need to spend, and the total cost invested in production during the uncertain situation and the normal period, the total expected quantity, many risk factors have been eliminated or minimized. Many researchers can use this prediction model to avoid the lack of attention to environmental measures and waste disposal. This suggested model is applicable to fish marketing, the production of dry fish, and added-value fish products. The wholesale and retail fisheries management face a number of planning difficulties during the uncertain fish ban period. The EOQ model of the fish market takes this into account and Verhulst’s demand for fish items follows an exponential distribution, which is a novel and important benefit of this research. The average total cost is decreased in the fuzzy arena in both the lockdown and regular circumstances by anticipating the backordering quantity. In order to increase sales, the demand for fish items during the difficult shut down should be determined and supplied to the retailers.

From society’s point of view, it is important to observe that, this prospective coordination model is of great benefit. As per the prerequisites of the system, this result can be developed for different types of demands to the added value fish products.

Future researchers can benefit from the current studies as well. The production setup with dual channels, application of type 1 and type 2 fuzzy and linguistics fuzzy concepts for uncertain parameters, can be examined in the next studies. The following are some of this study’s limitations:

Only two backlog cases have been addressed.

Only perishable products are included in this technique and total quantity and inventory level are increased steadily when this model is applied in fuzzy arena.

10 Compliance with ethical standards

10.1 Conflict of interest

The authors declare no conflict of interest.

10.2 Ethical approval

The article does not contain any studies with human participants or animals performed by the authors.

10.3 Informed consent

Informed consent was obtained from all individual participants included in the study.

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Economic ordering quantity inventory model with verhulst’s demand under fuzzy uncertainty for geographical market

Abstract

Keywords

1 Introduction

3 Basic definitions

3.1 Definition [20]

3.2 Definition [20]

3.3 Definition [20]

4 Assumption

5 Notations

7.1 Crisp model

7.2 Fuzzy mode

8 Sensitivity analysis

Table 2 Parameter changes in Backordered level % –30 –20 –10 10 20 30 E K 41.522 44.174 46.624 51.073 53.103 55.024 F c 40.151 43.263 46.171 51.522 54 56.392 t 47.713 48.122 48.522 49.304 49.681 50.044 H c 16 24 35 66 87.483 113 α 43.462 45.921 47.073 50.781 52.672 54.591

10 Compliance with ethical standards

10.1 Conflict of interest

10.2 Ethical approval

10.3 Informed consent

References

Table 2
Parameter changes in Backordered level

% –30 –20 –10 10 20 30

E _K 41.522 44.174 46.624 51.073 53.103 55.024

F _c 40.151 43.263 46.171 51.522 54 56.392

t 47.713 48.122 48.522 49.304 49.681 50.044

H _c 16 24 35 66 87.483 113

α 43.462 45.921 47.073 50.781 52.672 54.591