A fuzzy inventory model with three parameter Weibull deterioration with reliant holding cost and demand incorporating reliability

Abstract

This paper develops a fuzzy inventory model having price, stock, reliability and advertisement dependent demand with fuzzy holding cost varying on time and reliability. We also develop the inventory model for deteriorating items with three parameter weibull distribution pattern. Inflation has the reverse effect on the time value of money, so we consider the effect of inflation when there is shortage in the stock. The model also allows condition of permissible delay in payment. The credit period is less than or equal and greater than to the cycle time for settling the account which are the two cases for payment. The optimum solution of the inventory model are evaluated under crisp and fuzzy environments. We have used symmetric triangular fuzzy parameters for the proposed inventory model, and to defuzzify for the solution we use total λ-integral value. The model is illustrated with the help of numerical examples, and further sensitivity analysis of the decision variables has been done to examine the effect of changes in the values of the parameters on the optimal inventory policy under fuzziness.

Keywords

Shortage deterioration permissible delay in payment reliability advertisement triangular fuzzy number

1. Introduction

Deterioration of items in an inventory is a common phenomenon in real life situation or in business situations which makes the product value dull. Deterioration in each product cannot be completely avoided and the rate of deterioration for each product will vary due to the fact that the items in the inventory become obsolete, devalued, decayed or damaged depending on the type of goods. There are many items in real life that are subject to significant rate of deterioration. In case of durable products such as iron items, plastic gift, toys, steal items andhardware deterioration rate is very small, but in semi durable product, deterioration rate rapidly changes over the time such as fish, sweet, vegetable, chicken, bread, fruits, alcohol and milk products. In most of the cases, researchers considered two-parameter Weibull distribution for deterioration and non-instantaneous deteriorating items [1 –3] to develop an inventory model. Haider [4] constructed an inventory model with effect of deterioration and instantaneous replenishment with imperfect quality items. Tiwari et al. [5] developed an optimal pricing and lot-sizing policy for supply chain system with deteriorating items under limited storage capacity. Sarkar and Sarkar [6] proposed an inventory model with variable deterioration. Tiwari et al. [7] also proposed joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain. Mishra et al. [8] developed an inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment. Shabani et al. [9] proposed an inventory model with fuzzy deterioration rate under conditionally permissible delay in payments. Sanni and Chukwu [10] presented a three-parameter Weibull distribution deterioration to develop an inventory model. In this paper we propose an inventory model for deterioration as three-parameter Weibull distribution.

Now-a-days, a product is accepted in the society based on the reliability and also through the advertisement in the modern mass/ electronic/ print media. In the situation of present competitive market, the frequency of advertisement of an item changes its demand pattern among the public. The propaganda and canvassing of an item by time-to-time glamorous advertisements in the well-known media such as TV, radio, newspaper, magazine, etc. and also through the sales representatives have a motivational effect on the people to buy more. In traditional belief, it is generally assumed that the demand rate is independent of factors like stock availability, price of items, etc., during the sale period but it needs to be discussed. However, in actual practice observed that the demand for certain items is greatly influenced by the stock level, it seems to induce more consumers to buy it owing to its visibility, popularity or variety. Conversely, low stocks of certain goods might raise the perception that they are not fresh. Several researchers presented inventory model for price-dependent demand [11, 12], model with general demand [13], model for ramp type demand [14 –16], model with stock dependent demand [17 –23], model with stochastic demand [24, 25], model for inflation dependent rate under permissible delay in payments [26 –28]. Kundu et al. [29] developed an imperfect EPQ model for deteriorating items with promotional effort dependent demand. A very limited study has been incorporated on inventory model for reliability dependent demand [30, 31]. It would be more appropriate study on an inventory model considering the demand is stock, price, advertisement and reliability dependent.

Many inventory models were developed under the assumption that the holding cost is constant for the entire inventory cycle but in case of storage of deteriorating items such as food products for the longer duration these food products should be kept in more sophisticated storage facilities and services needed, and therefore these cases the holding cost should be assumed as a function of time. Various functions describing holding cost were considered by several researchers like, nonlinear holding cost [20, 32], variable holding cost [33]. Shekarian et al. [34] developed an economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning. In reality, if the reliability decreases with time, major customers are not interested to this product, so these products are kept in storage for long time, so the holding cost is increases with time. Most of the EOQ models were considered with various type of holding cost but in real life the assumption is not true in general. Therefore, the holding cost may vary with time as well as the reliability, and this paper present the holding cost is time and reliability dependent.

Due to the uncertainty of inventory parameters, the cost function becomes imprecise in nature, and here we have considered the cost function is fuzzy in nature. Several researchers work on fuzzy inventory model due to impreciseness of the cost and also other parameters [35]. Mahapatra et al. [36] presented inventory model with fuzzy coefficient of objective and constraint via parametric geometric programming. De and Goswami [37] developed an EOQ model with fuzzy inflation and fuzzy deterioration rate. Mahapatra et al. [38] introduced an EPQ model with imprecise space constraint based on intuitionistic fuzzy optimization technique. Kazemi et al. [39] presented a fuzzy EOQ inventory model with backorders incorporating human learning. Pal et al. [40] developed an EPQ model for a ramp type demand with Weibull deterioration under inflation in finite time horizon in crisp and fuzzy environment. Jain et al. [41] constructed a fuzzy imperfect production and repair inventory model with time dependent demand under inflationary conditions. Bhuiya and Chakraborty [42] proposed a fuzzy random EPQ model with fuzzy defective rates and fuzzy inspection errors.

The paper is organized as follows. In Section 2, we describe the assumptions and notations used throughout this paper. In Section 3, we describe three parameter Weibull distribution in inventory model. Section 4 present the mathematical formulation and analysis of the proposed model. In Section 5, we optimize the proposed inventory model for different cases. In Section 6, we establish inventory model under fuzzy environment. Section 7 demonstrate practical implications through numerical examples to illustrate the theorem and proposed analytical results. Section 8 exhibits a sensitivity analysis to study the effects of changes of the system parameters on the inventory model. Finally, conclusions are given in Section 9.

2. Assumptions and notation

Inventory model is developed under the following assumptions and notations.

2.1. Notations

I (t) inventory level at time t

r the reliability

p selling price

D (I (t); r, p) demand rate

q inflation rate

T₁ time when inventory level vanishes

T replenishment cycle

M credit period fixed by supplier to retailer

C_h holding cost per unit item

C_s shortage cost per unit item

C_d cost of each deteriorated item

C₁ purchase cost per unit item

A the frequency of advertisement

l_e rate of interest earned

l_p rate of interest due to delay in payment

2.2. Assumptions

Demand rate is dependent on the price factor k (p) = ηe^-pδ where η, δ > 0 are the parameters.

The demand is also dependent on stock dependent consumption rate parameter b where 0 ≤ b ≤ 1, reliability r and the frequency of advertisement A, the demand is D (I (t) ; r, p) = k (p) [ar^-c + bI (t)] A^ν where a, ν > 0.

Distribution of the time to deterioration follows a three-parameter Weibull distribution.

Shortages are allowed.

If the retailer pays within the offered credit period M not to pay any interest but has to pay interest at the rate I_p to the supplier if the retailer pays after credit period M.

The holding cost is dependent on time and reliability, i.e. h = tr^-t .

Fig.1

Time and rate of deterioration for 3-parameter Weibull distribution

Fig.2

Graphical representation of inventory system.

3. Development of three parameter Weibull distribution in inventory model

Deterioration rate vs time relation for three-parameter Weibull distribution is shown in Fig. 1, which shows that the three-parameter Weibull distribution is most appropriate for items with any initial value of the rate of deterioration and for items which start deteriorating only after a certain period of time. Sanni and Chukwu [10] developed an inventory model with three-parameter Weibull deterioration. Pal et al. [40] developed an EPQ model of ramp type demand with Weibull deterioration. With above assumptions, the on-hand inventory level at any instant of time is exhibited in Fig. 2.

Let us consider Q be the initial inventory level. The inventory level decreases due to deterioration and demand. Figure 2 shows that the length of the cycle T is divided into two sub-periods, where T₁ is the period of the system with on-hand inventory, and (T - T₁) is the period of the system with shortages. At the time T₁ the inventory level reaches zero and shortages occurring in the period (T₁, T) which is completely backlogged. The total number of backlogged items is replaced by the next replenishment. The distribution of the time to deterioration of the items follows a three-parameter Weibull distribution. The three-parameter Weibull density function is given by f (t) = αβ (t - γ) ^β-1e^{-α(t-γ)^β} where f (t) is probability density function, α > 0 is scale parameter, β > 0 is shape parameter, t > 0 is time of deterioration and γ is location parameter where t ≥ γ .

The instantaneous rate of deterioration of the non-deteriorated inventory at time t, Y (t) can be obtained from $Y (t) = \frac{f (t)}{1 - F (t)}$ , where F (t) is the distribution function for the three parameter Weibull distribution. Therefore, the instantaneous rate of deterioration of the on-hand inventory is Y (t) = αβ (t - γ) ^β-1. The probability density function represents the deterioration of items which is shown in Fig. 1, which is either decreasing, constant or increasing rate of deterioration. It is clear from Fig. 1 that the three-parameter Weibull distribution is appropriate for items with any initial value of the rate of deterioration, and for items which start deteriorating only after a certain period. To develop this mathematical model, we have derived cases under two different circumstances.

Case I: Length of period with positive stock, which is longer than the credit period M, i.e. M ≤ T₁ . In that case the buyer tries to earn interest at an annual rate I_e in [0, T₁] using the sales revenue. Beyond the credit period, the unsold is supplied to be financed with an annual rate I_p.

Case II: The permissible delay in payment M is greater than T₁ i.e. M ≥ T₁ . In that case the buyer pays no interest but earns interest at an annual rate I_e during the period (0, M).

4. Mathematical formulation and analysis of proposed inventory model

Let I (t) be the inventory level at time t (0 ≤ t ≤ T). The differential equations for the instantaneous state I (t) over (0, T) are given by

$\begin{matrix} \frac{dI (t)}{dt} + θ (t) I (t) \\ = - k (p) [{ar}^{- c} + bI (t)] A^{ν}, 0 \leq t \leq T_{1} \end{matrix}$ (1) $\frac{dI (t)}{dt} = - k (p) {ar}^{- c} A^{ν}, T_{1} \leq t \leq T$ (2) with the boundary conditions I (0) = Q, I (T₁) =0 .

The solution of above differential equations, after applying the boundary conditions I (0) = Q and I (T₁) =0 is $\begin{matrix} I (t) = \frac{R}{B} {e^{B (T_{1} - t)} - 1}, 0 \leq t \leq T_{1} \\ = R (T_{1} - t), T_{1} \leq t \leq T \end{matrix}$ (3) where R = aηe^-pδr^-cA^ν, B = αβ (- γ) ^β-1 + A^νηe^-pδb, and q_r = q + logr for simplify the mathematical expressions.

So, the maximum inventory is $I (0) = Q = \frac{R}{B} (e^{{BT}_{1}} - 1)$ and maximum shortage is I (T) = R (T₁ - T)

Now the inventory holding cost (HC) during [0, T₁] is

$= C_{h} \int_{0}^{T_{1}} {tr}^{- t} e^{- qt} I (t) dt$ $\begin{matrix} = \frac{{RC}_{h}}{B} [\frac{B {1 + q_{r} T_{1}} {B + 2 q_{r}}}{e^{{qT}_{1}} r^{T_{1}} q_{r}^{2} {(B + q_{r})}^{2}} \\ + \frac{e^{{BT}_{1}} - {BT}_{1} e^{- {qT}_{1}} r^{- T_{1}}}{{(B + q_{r})}^{2}} - \frac{1}{q_{r}^{2}}] \end{matrix}$ (4)

Now the shortage cost (SC) is $\begin{matrix} = - C_{s} \int_{T_{1}}^{T} I (t) e^{- qt} dt \\ = \frac{C_{S} R}{q^{2}} [e^{- {qT}_{1}} - e^{- qT} {1 + q (T - T_{1})}] \end{matrix}$ (5)

The purchase cost (PC) is

$= C_{1} Q + C_{1} e^{- qT} \int_{0}^{T - T_{1}} k (p) {ar}^{- c} A^{ν} dt$

$= C_{1} R {\frac{e^{{BT}_{1}} - 1}{B} + e^{- qT} (T - T_{1})}$ (6)

The Deterioration cost (DC) is

$= C_{d} [Q - \int_{0}^{T_{1}} D (I (t), r, p) dt]$

$\begin{matrix} = C_{d} R (1 - \frac{N}{B}) {\frac{1}{B} (e^{{BT}_{1}} - 1) - T_{1}} \end{matrix}$ (7) where N = A^νηbe^-pδ

Ordering cost (OC) is $OC = A_{0}$ (8)

Case 1: Period of positive stock (T₁) is longer than the credit period (M)

The interest earned per cycle (IE₁) is the interest earned during the positive inventory level, and is given by Interest earned in the time horizon up to T₁ is given by

${IE}_{1} = C_{1} I_{e} \int_{0}^{T_{1}} tD (I (t), r, p) dt$ $= {RC}_{1} I_{e} [(1 - \frac{N}{B}) \frac{T_{1}^{2}}{2} - \frac{N}{B^{3}} (1 + {BT}_{1} - e^{{BT}_{1}})]$ (9)

In this case, the credit time expires on or before the inventory depleted completely to zero. The interest payable (IP₁) per cycle for the inventory not being sold after the due date M is Interest payable in the time horizon.

${IP}_{1} = C_{1} I_{p} \int_{M}^{T} I (t) dt$

$= C_{1} I_{p} (\int_{M}^{T_{1}} I (t) dt + \int_{T_{1}}^{T} I (t) dt)$

$\begin{matrix} = {RC}_{1} I_{p} {\frac{(M - T_{1})}{B} - \frac{{(T - T_{1})}^{2}}{2} + \frac{e^{B (T_{1} - M)} - 1}{B^{2}}} \end{matrix}$ (10)

The total variable cost is comprised of the ordering cost(OC), shortage cost(SC), purchase cost(PC), holding cost(HC), deterioration cost(DC) and interest payable(IP₁) minus the interest earned(IE₁). i.e. the total cost per unit item per unit time is

${TC}_{1} = \frac{1}{T} (OC + SC + PC + DC + HC + {IP}_{1} - {IE}_{1})$ $\begin{matrix} = \frac{1}{T} [A_{0} + \frac{C_{S} R}{q^{2}} {e^{- {qT}_{1}} - e^{- qT} (1 + q (T - T_{1}))}] \\ + \frac{1}{T} C_{1} R {\frac{(e^{{BT}_{1}} - 1)}{B} + e^{- qT} (T - T_{1})} \\ + \frac{C_{h} R}{TB} {\frac{B (1 + q_{r} T_{1}) (B + 2 q_{r} T_{1})}{e^{{qT}_{1}} r^{T_{1}} q_{r}^{2} (B + q_{r})^{2}} \\ + \frac{e^{{BT}_{1}} - {BT}_{1} e^{- {qT}_{1}} r^{- T_{1}}}{{(B + q_{r})}^{2}} - \frac{1}{q_{r}^{2}}} \\ + \frac{1}{T} C_{d} R (1 - \frac{N}{B}) {\frac{1}{B} (e^{{BT}_{1}} - 1) - T_{1}} \\ - \frac{1}{T} {RC}_{1} I_{e} {(1 - \frac{N}{B}) \frac{T_{1}^{2}}{2} - \frac{N}{B^{3}} (1 + {BT}_{1} - e^{{BT}_{1}})} \\ + \frac{1}{T} {RC}_{1} I_{p} {\frac{M - T_{1}}{B} - \frac{{(T - T_{1})}^{2}}{2} + \frac{e^{B (T_{1} - M)} - 1}{B^{2}}} \end{matrix}$ (11)

Case 2: Permissible delay in payment (M) is greater than credit period (T₁)

Interest earned (IE₂) due to arrival of supplier after the stock ends. $\begin{matrix} {IE}_{2} & = & C_{1} I_{e} [\int_{0}^{T_{1}} tD (I (t), r, p) dt \\ + (M - T_{1}) \int_{0}^{T_{1}} D (I (t), r, p) dt] \end{matrix}$

$\begin{matrix} = {RC}_{1} I_{e} [T_{1} M - (1 + \frac{N}{B}) \frac{T_{1}^{2}}{2}] \\ - {RC}_{1} I_{e} [\frac{N (1 + {BT}_{1} - e^{{BT}_{1}}) {1 + B (M - T_{1})}}{B^{3}}] \end{matrix}$ (12)

Here the retailer has sold Q unit during [0, T₁] and is paying C₁Q to the supplier, so the retailer does not have to pay any interest because the supplier can be paid in full at time M, so interest charge is zero. ${IP}_{2} = 0$ (13)

The total variable cost is comprised of the sum of the ordering cost(OC), shortage cost(SC), purchase cost(PC), holding cost(HC) and deterioration cost(DC) minus the interest earned(IE₁). i.e. the total cost per unit item per unit time is

${TC}_{2} = \frac{1}{T} (OC + HC + SC + PC + DC + {IP}_{2} - {IE}_{2})$ $\begin{matrix} = \frac{1}{T} [A_{0} + \frac{C_{S} R}{q^{2}} {\frac{1}{e_{1}^{qT}} - \frac{(1 + q (T - T_{1}))}{e^{qT}}}] \\ + \frac{C_{1} R}{T} {\frac{(e^{{BT}_{1}} - 1)}{B} + \frac{T - T_{1}}{e^{qT}}} \\ + \frac{C_{d} R}{T} (1 - \frac{N}{B}) {- T_{1} + \frac{(e^{{BT}_{1}} - 1)}{B}} \\ + \frac{C_{h} R}{TB} {\frac{B {1 + q_{r} T_{1}} {B + 2 q_{r}}}{e^{{qT}_{1}} r^{T_{1}} q_{r}^{2} {(B + q_{r})}^{2}} \\ + \frac{e^{{BT}_{1}} - {BT}_{1} e^{- {qT}_{1}} r^{- T_{1}}}{{(B + q_{r})}^{2}} - \frac{1}{q_{r}^{2}}} \\ - \frac{{RC}_{1} I_{e}}{T} {T_{1} M - (1 + \frac{N}{B}) \frac{T_{1}^{2}}{2}} \\ + \frac{{RC}_{1} I_{e}}{T} {\frac{N (1 + {BT}_{1} - e^{{BT}_{1}}) {1 + B (M - T_{1})}}{B^{3}}} \end{matrix}$ (14)

5. Optimization of proposed inventory model for different cases

In management point of view the production or manufacturing cost should be optimum and hence our objective is to minimize the total cost per unit time. So, optimizing the proposed model by as usual differentiation method.

Case I: The necessary and sufficient condition to minimize TC₁ for a given value of M is respectively $\frac{{dTC}_{1}}{{dT}_{1}} = 0$ and $\frac{d^{2} {TC}_{1}}{{dT}_{1}^{2}} > 0 .$

From 11 it follows that

$\begin{matrix} \frac{{dTC}_{1}}{{dT}_{1}} = \frac{R}{T} [\frac{C_{S}}{q} (e^{- qT} - e^{- {qT}_{1}}) + C_{1} (e^{{BT}_{1}} - e^{- qT})] \\ + \frac{{RC}_{d}}{T} (1 - \frac{N}{B}) (e^{{BT}_{1}} - 1) \\ + \frac{C_{h} R {e^{{BT}_{1}} - e^{- {qT}_{1}} r^{- T_{1}} (1 + T_{1} (B + q_{r}))}}{T {(B + q_{r})}^{2}} \\ + \frac{C_{1} I_{p} R}{BT} {e^{B (T_{1} - M)} - (1 + (T_{1} - T) B)} \\ - \frac{C_{1} I_{e} R}{T} {(1 - \frac{N}{B}) T_{1} - \frac{N}{B^{2}} (1 - e^{{BT}_{1}})} \end{matrix}$ (15)

The optimum value of $T_{1} = t_{1}^{*}$ provided it satisfies the condition given below

$\begin{matrix} \frac{C_{h} {Be}^{{BT}_{1}}}{{(B + q_{r})}^{2}} + \frac{C_{h} e^{- {qT}_{1}} r^{- T_{1}}}{{(B + q_{r})}^{2}} q_{r} + \frac{C_{h} e^{- {qT}_{1}} r^{- T_{1}}}{(B + q_{r})} (T_{1} q_{r} - 1) \\ + C_{S} e^{- {qT}_{1}} + C_{1} {Be}^{{BT}_{1}} + C_{d} (B - N) e^{{BT}_{1}} \\ + C_{1} I_{P} (e^{B (T_{1} - M)} - 1) - C_{1} T_{e} {1 + \frac{N}{B} (e^{{BT}_{1}} - 1)} > 0 \end{matrix}$ (16)

Case II: The necessary and sufficient condition to minimize TC₂ for a given value of M is respectively $\frac{{dTC}_{2}}{{dT}_{1}} = 0$ and $\frac{d^{2} {TC}_{2}}{{dT}_{1}^{2}} > 0 .$

From Eq15 it follows that $\begin{matrix} \frac{{dTC}_{2}}{{dT}_{1}} = \frac{R}{T} [\frac{C_{S}}{q} (e^{- qT} - e^{- {qT}_{1}}) + C_{1} (e^{{BT}_{1}} - e^{- qT})] \\ + \frac{{RC}_{d}}{T} (1 - \frac{N}{B}) (e^{{BT}_{1}} - 1) \\ + \frac{C_{h} R {e^{{BT}_{1}} - e^{- {qT}_{1}} r^{- T_{1}} (1 + T_{1} (B + q_{r}))}}{T {(B + q_{r})}^{2}} \\ - \frac{C_{1} I_{e} R}{T} (M - T_{1}) {1 - \frac{N}{B} (1 - e^{{BT}_{1}})} \end{matrix}$ (17)

The optimum value of $T_{1} = t_{2}^{*}$ provided it satisfies the condition given below $\begin{matrix} \frac{C_{h} {Be}^{{BT}_{1}}}{{(B + q_{r})}^{2}} + \frac{C_{h} e^{- {qT}_{1}} r^{- T_{1}}}{{(B + q_{r})}^{2}} q_{r} + \frac{C_{h} e^{- {qT}_{1}} r^{- T_{1}}}{(B + q_{r})} (T_{1} q_{r} - 1) \\ + C_{S} e^{- {qT}_{1}} + C_{1} {Be}^{{BT}_{1}} + C_{d} (B - N) e^{{BT}_{1}} \\ + C_{1} I_{e} {1 - \frac{N}{B} + {Ne}^{B T_{1}} (\frac{1}{B} - M + T_{1})} > 0 \end{matrix}$ (18)

6. Proposed inventory model under fuzzyness

Fuzzy set [43] is providing a new mathematical tool which helps us to handle imprecise and ambiguous notion in formulation and best analysis of practical problems.

Fuzzy Set: A fuzzy set $\tilde{A}$ is defined by a set or ordered pairs, a binary relation, $\tilde{A} = {(x, μ_{\tilde{A}} (x) : x \in A, μ_{\tilde{A}} (x) \in [0, 1]}$ where $μ_{\tilde{A}} (x)$ is called the membership function, which specifies the grade or degree of any element x in A belongs to the fuzzy set $\tilde{A}$ .

Triangular Fuzzy Number: A triangular fuzzy number (TFN) or simply triangular number $\tilde{A} = (a_{1}, a_{2}, a_{3})$ has three parameters a₁, a₂, a₃ where a₁ < a₂ < a₃ with membership function $μ_{\tilde{A}} (x)$ is defined by

$μ_{\tilde{A}} (x) = {\begin{matrix} μ_{L \tilde{A}} (x) = \frac{x - a_{1}}{a_{2} - a_{1}} if a_{1} \leq x \leq a_{2} \\ μ_{R \tilde{A}} (x) = \frac{a_{3} - x}{a_{3} - a_{2}} if a_{2} \leq x \leq a_{3} \\ 0 otherwise \end{matrix}$ (19)

A TFN $\tilde{A} = (a_{1}, a_{2}, a_{3})$ is represented pictorially as follows

Fig.3

Triangular fuzzy number

Graded Mean Value: Let λ be a pre-assigned parameter also called the degree of optimism and λ ∈[0, 1]. The graded mean value or the total λ integral value [35] of $\tilde{A}$ is defined as $I_{λ} (\tilde{A}) = λ I_{R} (\tilde{A}) + (1 - λ) I_{L} (\tilde{A})$ where $I_{R} (\tilde{A})$ and $I_{L} (\tilde{A})$ are the left and right interval values of $\tilde{A}$ and it is defined as

$\begin{matrix} I_{L} (\tilde{A}) & = & \int_{0}^{1} (μ_{L \tilde{A}})^{- 1} α d α and I_{R} (\tilde{A}) \\ = & \int_{0}^{1} (μ_{R \tilde{A}})^{- 1} α d α \end{matrix}$

For TFN $\tilde{A} = (a_{1}, a_{2}, a_{3}),$ we have $(μ_{L \tilde{A}})^{- 1} α = a_{1} + α (a_{2} - a_{1})$ and $(μ_{R \tilde{A}})^{- 1} α = a_{3} - α (a_{3} - a_{2})$ .

Therefore the left and right integral values are $I_{L} (\tilde{A}) = \frac{a_{1} + a_{2}}{2}$ and $I_{R} (\tilde{A}) = \frac{a_{2} + a_{3}}{2} .$ Hence the graded mean value or total λ integral value of $\tilde{A}$ is $I_{λ} (\tilde{A}) = λ (\frac{a_{2} + a_{3}}{2}) + (1 - λ) (\frac{a_{1} + a_{2}}{2}) .$

This paper propose selling price, holding cost and inflation rate are fuzzy, since due to various reasons in the market as well as in production place, the above rate and costs fluctuate with time. So we have tried to modify the proposed inventory model by introduce fuzzy for the above rate and costs to make it coincide with the real situation and check the effect on proposed inventory model.

The holding cost C_h is replaced by TFN ${\tilde{C}}_{h} =$ (C_h - δ₁, C_h, C_h + δ₁) where 0 < δ₁ < C_h . Here we consider a₁ = C_h - δ₁, a₂ = C_h, a₃ = C_h + δ₁ . Hence the graded mean value (or total λ integral value) of ${\tilde{C}}_{h}$ is $I_{λ} ({\tilde{C}}_{h}) = λ (C_{h} + \frac{δ_{1}}{2}) + (1 - λ) (C_{h} - \frac{δ_{1}}{2}) = C_{h} + (λ - \frac{1}{2}) δ_{1} .$

The selling price p is represent by TFN $\tilde{p} = (p - δ_{2}, p, p + δ_{2})$ where 0 < δ₂ < p and in similar way we get $I_{λ} (\tilde{p}) = p + (λ - \frac{1}{2}) δ_{2} .$

The inflation rate q is represent by TFN $\tilde{q} = (q - δ_{3}, q, q + δ_{3})$ where 0 < δ₃ < q and we get $I_{λ} (\tilde{q}) = q + (λ - \frac{1}{2}) δ_{3} .$

The fuzzy parameters ${\tilde{C}}_{h},$ $\tilde{p}$ and $\tilde{q}$ can be used on the inventory model and hence the total costs TC is also become fuzzy in nature and this determines the fuzzy total expected cost ( $\tilde{TC}$ ) per unit time.

Case 1: Period of positive stock (T₁) is longer than credit period M i.e. M ≤ T₁

$\begin{matrix} \tilde{TC} = \frac{1}{T} [A_{0} + \frac{C_{S} R {e^{- q_{δ} T_{1}} - e^{- q_{δ} T} (1 + q_{δ} (T - T_{1}))}}{q_{δ}^{2}} + C_{1} R {\frac{(e^{{BT}_{1} - 1})}{B} + (T - T_{1}) e^{- q_{δ} T}} \\ + \frac{(C_{h} + (λ - \frac{1}{2}) δ_{1}) R}{B} {\frac{B {1 + q_{δ_{1}} T_{1}} {B + 2 q_{δ_{1}}}}{e^{q_{δ_{1}}} r_{1}^{T} q_{δ_{1}}^{2} {(B + q_{δ_{1}})}^{2}} + \frac{e^{{BT}_{1}} - {BT}_{1} e^{- q_{δ} T_{1}} r^{- T_{1}}}{{(B + q_{δ_{1}})}^{2}} - \frac{1}{(q_{δ_{1}})^{2}}} {- RC}_{1} I_{e} {(1 - \frac{N}{B}) \frac{T_{1}^{2}}{2} - \frac{N}{B^{3}} (1 + {BT}_{1} - e^{{BT}_{1}})} \\ {+ RC}_{1} I_{p} {\frac{({M - T}_{1})}{B} - \frac{{({T - T}_{1})}^{2}}{2} + \frac{e^{B (T_{1} - M)} - 1}{B^{2}}} + C_{d} R (1 - \frac{N}{B}) {\frac{1}{B} (e^{{BT}_{1}} - 1) - T_{1}}] \end{matrix}$ (20)

where $R = a η e^{- (p + (λ - \frac{1}{2}) δ_{2}) δ} r^{- c} A^{ν}, B = α β$ $(- γ)^{β - 1} + A^{ν} η e^{- (p + (λ - \frac{1}{2}) δ_{2}) δ} b$ , q_δ = q+ $(λ - \frac{1}{2}) δ_{3}$ , and q_{δ
₁} = q_δ + logr

Case 2: Permissible delay in payment is greater than credit period i.e. M ≥ T₁

$\begin{matrix} \tilde{TC} = \frac{1}{T} [A_{0} + \frac{C_{S} R}{q_{δ}^{2}} {\frac{1}{e^{q_{δ} T_{1}}} - \frac{(1 + q_{δ} (T - T_{1}))}{e^{q_{δ} T}}} - {RC}_{1} I_{e} {T_{1} M - (1 + \frac{N}{B}) \frac{T_{1}^{2}}{2} - \frac{N (1 + {BT}_{1} - e^{{BT}_{1}}) {1 + B (M - T_{1})}}{B^{3}}} \\ + C_{1} R {\frac{(e^{{BT}_{1}} - 1)}{B} + \frac{(T - T_{1})}{e^{q_{δ} T}}} + \frac{(C_{h} + (λ - \frac{1}{2}) δ_{1}) R}{B} {\frac{B {1 + q_{δ_{1}} T_{1}} {B + 2 q_{δ_{1}}}}{e^{q_{δ} T_{1}} r^{T_{1}} q_{δ_{1}}^{2} {(B + q_{δ_{1}})}^{2}} + \frac{e^{{BT}_{1} - {BT}_{1} e^{- q_{δ} T_{1}}} r^{- T_{1}}}{{(B + q_{δ_{1}})}^{2}} - \frac{1}{q_{δ_{1}}^{2}}} \\ + C_{d} R (1 - \frac{N}{B}) {- T_{1} + \frac{(e^{{BT}_{1}} - 1)}{B}}] \end{matrix}$ (21)

Solving (20) and (21) we get the optimal total cost under fuzzyness. We obtain the optimal time and optimal value of the total inventory cost for the degree of optimism by changing the values of the parameter λ .

7. Numerical example with practical implications

Now-a-days, due to the competition, different multinational companies try to motivate customers by attractive advertisement in TV, newspapers, periodicals repeatedly also through sales representatives. They estimate the requirements (Demands) of their products by these advertisements from the experience of salespersons and with the help of experts’ opinion. In reality, there are so many physical goods, which deteriorate over time due to different factors like dryness, damage, spoilage, vaporization, etc., during the stock-in period. This deterioration effect depends on the preserving facilities of the warehouses. Hence, in the study of inventory problems, this effect cannot be ignored.

The fact is that the selling price, stock price, reliability and advertisement of an item can affect significantly with the demand of an item. Selling price, stock price, reliability and advertisement are the main criterion of the consumer to buy a particular item. The principal feature of the model is the deterministic demand rate which is assumed as a function of selling price, stock price, reliability and advertisement. Here, the holding cost and selling price optimize the net profit which are simultaneously optimized. In many realistic situations, stock out is unavoidable due to various uncertainties. There are many situations, where the profit of the stored item is higher than its back ordered cost. Hence, consideration of shortages is economically desirable in these cases.

We present an example to illustrate the results for the proposed model through numerical presentation under crisp and fuzzy environments for two different scenarios of ordering policies: Scenario-I: Payment before total depletion. Scenario-II: Payment after total depletion.

Example: In the supermarket, the demand rate not only depend upon the amount of the stock but also depends upon the reliability as well as the price of the item so that demand rate D (I (t) ; r, p) = k (p) [ar^-c + bI (t)] A^ν, here η = 250, δ = 1.9, a = 250, r = 0.7, c = 0.8, α = 0.06, b = 0.1, β = 1, A = 2, T = 1, ν = 0.7, γ = 1.

Let the cost price(C₁), selling price (p), holding cost (C_h), shortage cost (C_s) and cost of each deteriorate item (C_d) of each item be $2.5, $3, $3.7, $4.5 and $1.2 respectively. It takes $250 to order the total inventory. The system considers under inflation rate (q) of 5% and let the retailer earn (I_e) 15% of interest and pays (I_p) 18% interest where the total inventory system is considered for a full year.

Substituting all the values in (Eq12) and (Eq15) and we get the optimal value and the optimum total cost. Table-1 represent optimal average cost for different delay in payment in both the scenario. Let the supplier comes (i) Monthly (ii) Six months after (iii) Nine months after (iv) After one year.

Table 1
Optimal average cost for different delay in payment

Delay of Payment $T_{1} = t_{1}^{}$ ${TC}_{1}^{} (t_{1}^{})$ $T_{1} = t_{2}^{}$ ${TC}_{2}^{} (t_{2}^{})$

(year) ($) (year) ($)

1 month 0.6365 1607.2 0.6307 1603.9

(M = 0.083)

6 months 0.6626 1575.7 0.6508 1553.5

(M = 0.5)

9 months 0.6771 1573.3 0.6628 1522.6

(M = 0.75)

12 months 0.6908 1582.8 0.6747 1491.0

(M = 1)

Delay of Payment	$T_{1} = t_{1}^{*}$	${TC}_{1}^{} (t_{1}^{})$	$T_{1} = t_{2}^{*}$	${TC}_{2}^{} (t_{2}^{})$
1 month	0.6365	1607.2	0.6307	1603.9
(M = 0.083)
6 months	0.6626	1575.7	0.6508	1553.5
(M = 0.5)
9 months	0.6771	1573.3	0.6628	1522.6
(M = 0.75)
12 months	0.6908	1582.8	0.6747	1491.0
(M = 1)

Table 2

Optimal solution of the model in fuzzy environment

λ	M	$t_{1}^{*}$	${TC}_{1}^{} (t_{1}^{})$	$t_{2}^{*}$	${TC}_{2}^{} (t_{2}^{})$
		(year)	($)	(year)	($)
1	0.0833	0.6269	1369.6	0.6224	1405.2
0.5	0.6523	1344.4	0.6420	1366.9
0.75	0.6666	1342.9	0.6536	1343.3
1	0.6801	1351.2	0.6652	1319.3
0.7	0.0833	0.6328	1506.6	0.6275	1546.7
0.5	0.6585	1477.7	0.6474	1503.2
0.75	0.6729	1475.7	0.6592	1476.4
1	0.6866	1484.8	0.6710	1449.2
0.5	0.0833	0.6365	1607.2	0.6307	1650.7
0.5	0.6626	1575.7	0.6508	1603.4
0.75	0.6771	1573.3	0.6628	1574.2
1	0.6908	1582.8	0.6747	1544.6
0.2	0.0833	0.6420	1773.9	0.6352	1823.0
0.5	0.6684	1737.8	0.6557	1769.2
0.75	0.6831	1734.8	0.6680	1736.1
1	0.6969	1745.1	0.6801	1702.4
0	0.0833	0.6454	1896.5	0.6381	1949.7
0.5	0.6721	1857.0	0.6588	1891.2
0.75	0.6869	1853.6	0.6712	1855.2
1	0.7008	1864.5	0.6835	1818.5

Form the above table we observe that initially when delay of payment i.e. for one month (i.e., M = 0.0833) we see that t₁ > M and t₂ > M so here case-II is contradicting and only case- I hold and minimum average cost TC₁ . Again, when payment to supplier after six months (i.e., M = 0.5) inventory model follows the fact t₁ > M and t₂ > M so here also case-II is contradicting and only case- I hold and minimum average cost TC₁. For third case, i.e. payment to the supplier after nine months (i.e., M = 0.75) inventory model follows the fact t₁ < M and t₂ < M so here case- I is contradicting and only case- II hold and minimum average cost TC₂. Finally, for the last case i.e. payment to the supplier after four months (i.e., M = 1) inventory model follows the fact t₁ < M and t₂ < M so here also case- I is contradicting and only case- II hold and minimum average cost TC₂ .

The Table 2 presents the numerical solution of fuzzy inventory model by considering the fuzzy holding cost $C_{\hat{h}} =$ (C_h - δ₁, C_h, C_h + δ₁), selling price $\tilde{p} = (p - δ_{2}, p, p + δ_{2})$ , and inflation rate $\tilde{q} = (q - δ_{3}, q, q + δ_{3}),$ where values of δ₁ = 0.75, δ₂ = 0.2, δ₃ = 0.02 .The effect of different values of λ, such as 1 (Optimistic), 0.7 (About optimistic), 0.5 (Moderate), 0.2 (About pessimistic) and 0 (pessimistic) cases for finding the optimal cost.

Table 2 shows that with the effect of different values of λ, the total cost decreases upto M = 0.75 and then increases. Also the value of total cost increases as the value of λ decreases. We also observe that if we consider one fixed M then the total cost increases as we move from a optimistic point of view to pessimistic point of view. The model is realistic in fuzzy environment since in reality the selling price, holding cost and inflation rates are varying and indeterministic in nature.

8. Sensitivity analysis of parameters of proposed model

The sensitivity analysis for the parameters having crisp value, the effects of changes of values of the parameters C_d, C_h, C₁, C_s, p, q, r, ν, b, α, η, and I_e on the optimal total cost is presented. The sensitivity analysis is performed by changing each of parameters by -20 % , -10 % , 10% and 20% taking one parameter at a time and keeping the remaining parameters unchanged. Table 3 represent sensitive analysis for cost parameter of the proposed model for M = 0.0833.

Fig.4

Effect of optimal total cost for purchase cost C₁

Fig.5

Effect of optimal total cost for shortage cost C_s

Table 3

Sensitivity analysis of proposed model for cost parameters when M=0.0833

Parameters		Payment before credit period				Payment after credit period
Name	Change(%)	$t_{1}^{*}$	Change(%)	${TC}_{1} {(t}_{1}^{*})$	Change(%)	$t_{2}^{*}$	Change(%)	${TC}_{2} {(t}_{2}^{*})$	Change (%)
C _d	-20	0.6378	0.204	1605.9	-0.081	0.6318	0.174	1602.4	-0.094
	-10	0.6372	0.110	1606.6	-0.037	0.6313	0.095	1603.2	-0.044
	10	0.6359	-0.094	1607.9	0.044	0.6301	-0.095	1604.7	0.050
	20	0.6353	-0.189	1608.6	0.087	0.6295	-0.190	1605.5	0.100
C _h	-20	0.6615	3.928	1589.9	-1.076	0.6528	3.504	1583.8	-1.253
	-10	0.6486	1.901	1598.8	-0.523	0.6414	1.697	1594.2	-0.605
	10	0.6253	-1.760	1615.2	0.498	0.6207	-1.586	1613.2	0.580
	20	0.6148	-3.409	1622.7	0.964	0.6112	-3.092	1621.9	1.122
C ₁	-20	0.6530	2.592	1378.8	-14.211	0.6470	2.584	1374.5	-14.303
	-10	0.6448	1.304	1493.1	-7.099	0.6388	1.284	1489.2	-7.151
	10	0.6283	-1.288	1721.1	7.087	0.6228	-1.253	1718.6	7.151
	20	0.6201	-2.577	1834.8	14.161	0.6150	-2.489	1833.2	14.296
C _s	-20	0.5894	-7.400	1578.2	-1.804	0.5878	-6.802	1580.5	-1.459
	-10	0.6147	-3.425	1593.6	-0.846	0.6106	-3.187	1592.9	-0.686
	10	0.6558	3.032	1619.4 0.759	0.6486	2.838	1613.9	0.623
	20	0.6728	5.703	1630.4	1.444	0.6646	5.375	1622.9	1.185

Table 4

Sensitivity analysis of proposed model for other than cost parameter when M=0.0833

Parameters		Payment before credit period				Payment after credit period
Name	Change(%)	$t_{1}^{*}$	Change(%)	${TC}_{1} {(t}_{1}^{*})$	Change(%)	$t_{2}^{*}$	Change(%)	${TC}_{2} {(t}_{2}^{*})$	Change(%)
	-20	0.5649	-11.249	4713.6	193.280	0.5617	-10.940	4710.6	193.697
p	-10	0.6098	-4.195	2697.1	67.814	0.6049	-4.091	2687.7	67.573
	10	0.6520	2.435	1008.7	-37.239	0.6457	2.378	1010.5	-36.997
	20	0.6609	3.833	676.2	-57.927	0.6543	3.742	680.1	-57.597
q	-20	0.6408	0.676	1612.5	0.330	0.6346	0.618	1608.7	0.299
	-10	0.6387	0.346	1609.9	0.168	0.6326	0.301	1606.3	0.150
	10	0.6344	-0.330	1604.6	-0.162	0.6287	-0.317	1601.5	-0.150
	20	0.6322	-0.676	1601.9	-0.330	0.6268	-0.618	1599.1	-0.299
r	-20	0.6255	-1.728	1879.6	16.949	0.6209	-1.554	1877.1	17.033
	-10	0.6313	-0.817	1729.6	7.616	0.6261	-0.729	1726.7	7.656
	10	0.6412	0.738	1505.3	-6.340	0.6348	0.650	1501.7	-6.372
	20	0.6455	1.414	1418.9	-11.716	0.6386	1.253	1415.2	-11.765
I _e	-20	0.6301	-1.006	1614.2	0.436	0.6360	0.840	1607.3	0.212
	-10	0.6333	-0.503	1610.8	0.224	0.6333	0.412	1605.6	0.106
	10	0.6398	0.518	1603.7	-0.218	0.6280	-0.428	1602.3	-0.100
	20	0.6431	1.037	1600.1	-0.442	0.6254	-0.840	1600.8	-0.193
ν	-20	0.6398	0.518	1478.7	-7.995	0.6339	0.507	1476.5	-7.943
	-10	0.6382	0.267	1541.3	-4.100	0.6323	0.254	1538.6	-4.071
	10	0.6348	-0.267	1676.6	4.318	0.6290	-0.270	1672.7	4.290
	20	0.6329	-0.566	1749.5	8.854	0.6272	-0.555	1745.1	8.804
b	-20	0.6436	1.115	1600.1	-0.442	0.6376	1.094	1598.9	-0.312
	-10	0.6401	0.566	1603.7	-0.218	0.6341	0.539	1601.3	-0.162
	10	0.6330	-0.550	1610.8	0.224	0.6273	-0.539	1606.8	0.181
	20	0.6295	-1.100	1614.3	0.442	0.6239	-1.078	1609.8	0.368
α	-20	0.6411	0.723	1602.6	-0.286	0.6347	0.634	1595.8	-0.505
	-10	0.6388	0.361	1604.9	-0.143	0.6327	0.317	1599.9	-0.249
	10	0.6343	-0.346	1609.5	0.143	0.6287	-0.317	1607.8	0.243
	20	0.6320	-0.707	1611.8	0.286	0.6267	-0.634	1611.6	0.480
η	-20	0.6436	1.115	1330.1	-17.241	0.6376	1.094	1329.1	-17.133
	-10	0.6401	0.566	1468.3	-8.642	0.6341	0.539	1466.2	-8.585
	10	0.6330	-0.550	1746.9	8.692	0.6273	-0.539	1742.5	8.641
	20	0.6295	-1.100	1887.2	17.422	0.6239	-1.078	1881.7	17.320

Fig.6

Effect of optimal total cost for selling price p

Fig.7

Effect of optimal total cost for reliability r

Fig.8

Effect of optimal total cost for shape parameter ν

Fig.9

Effect of optimal total cost for shape parameter η

On the basis of sensitivity analysis which is obtained in the above Table 3, the model is less sensitive to deterioration cost (C_d), as C_d increases the total cost increases and time t₁ decreases. Holding for longer time, the commodities such as fruits, vegetables, foodstuffs are subject to direct spoilage while kept in store. Highly volatile liquids such as gasoline, alcohol, turpentine, undergo physical depletion over time through the process of evaporation. Electronic goods, radioactive substances, photographic film, grain, deteriorate through a gradual loss of potential or utility with the passage of time. The model is less sensitive to holding cost (C_h) i.e., if there is increase the holding cost the total cost increases and t₁ will decrease. It means, if we hold for longer time with high value of holding cost then obviously the total cost will increase when our aim is to reduce the total cost. Thus, with the increase in holding cost the replenishment time decreases i.e., we have to hold the item for less time, in order to obtain the minimum total cost.

The model is highly sensitive to the purchase cost (C₁) and the shortage cost (C_s). If there is increase the purchase cost the total cost increases and t₁ will also decrease. For increasing the shortage cost the total cost increases and t₁ will also increase, i.e. if the shortage cost increases then the total cost will also increase and to obtain an optimum total cost, time will increase because if the time t₁ increased by some percentage then the shortage will start at that time and the shortage period will be decreased. Figure 4 and 5 gives a graphical representation of the percentage change in total cost with change of time and change of C₁ and C_s in four cases of M respectively.

On the basis of sensitivity analysis which is obtained in the above Table 4, it is observed that the model is very highly sensitive to selling price (p), highly sensitive to reliability parameter (r) and shape parameter ν, η and less sensitive to inflation rate parameter (q), rate of interest earned (I_e), and shape parameter α, b . As price p increases by some percentage, the total cost will decrease and time t₁ will increase. It means that if we increase the selling price the value of money has decreased and hence the total cost has decreased. The reliability parameter (r) increases by some percentage the total cost will decreases and time t₁ will increase, since the reliability parameter increases means the demand will automatically increase and if demand increases means it holds for less time, hence the total cost will decrease. Figure 6 and 7 gives a graphical representation of the percentage change in total cost with change of time respectively for the change of p and r in all cases of M.

The model is highly sensitive to ν, since increase of this the shape parameter means advertisement increases and so the demand will increase and time will decrease simultaneously, hence the total cost will increase. The model is also highly sensitive to η, because this shape parameter increases means price factor increases, and so the demand will increase and time will decrease simultaneously. Figure 8 and 9 gives a graphical representation of the percentage change in total cost with change of time and change of shape parameter (ν) and (η) respectively in all cases of M.

The model is less sensitive to rate of interest earned (I_e), reason for this when I_e rate has decreased by some percentage so the retailer is earning less which leads to increase in costing of the total inventory system. The model is less sensitive to shape parameter (α), as α increases the deterioration rate will increases and hence to obtain an optimum total time will decrease and total cost will increase. The model is also less sensitive to inflation rate (q) because the inflation rate has increased means the demand has increased and hence the total cost will increase. The proposed model is less sensitive to shape parameter (b), as b increases means demand increases and time decreases simultaneously.

9. Conclusion

In this paper, we have developed an inventory model with time and reliability dependent holding cost with price, stock, reliability and advertisement dependent demand and three parameter Weibull deterioration and delay in payment. The principal feature of the model is the demand rate which is assumed as a function of selling price, stock, reliability and advertisement. Therefore, the problems of determining the retail price and lot-size are inter-dependent. It should be noted that in order to maximize the profit, a vendor can either increase the price or inflation or shorten the replenishment cycle or increases the reliability or advertisement or shorten the inventory holding time to counteract a greater loss due to a higher deterioration rate.

This paper discussed the inventory model with fuzzy parameters and observed that the total cost in the deterministic case is the same as that of which reflects a moderately optimistic decision-maker’s view point for fuzzy model. The sensitivity analysis shows that the model is sensitive with the change in parameter as per the formulation of the proposed fuzzy model. The changes of purchase cost, price of the stock and reliability are more effective in the proposed inventory model with the huge changes in total cost and also inflation and advertisement are effective in the proposed inventory model with the changes in total cost. It helps retailer to make decisions in different replenishment policies. This study is presented to determine at what time should the stock ends so that the total cost of storage is minimum. This model can give different decision based on the choice of decision maker due to the study of fuzziness of the parameters of the proposed inventory model.

Footnotes

Acknowledgement

We are grateful to Editor-in-chief, Subject Editor and anonymous Referees for their valuable comments and helpful suggestions which have helped us to improve this work significantly.

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