The main purpose of this paper is to develop an inventory model under fuzzy approach by considering the effect of inflation and time value of money, to determin e the optimal time period for inventory cycle and minimum total average costs. The model is integrated production inventory model developed where; the Demand has a direct linear impact on production rate. The model can be divided into four stages. In the first two stages with original production rate and subsequent change in production rate, inventory level rises. Third stage is time after the accumulation of inventory and before the deterioration starts, where demand which selling price dependent is depreciating the inventory level, while in the fourth stage deterioration occurs, which is consider ed to follow two parameter Weibull distribution. The back-order is not considered. Hexagonal fuzzy numbers are used to derive optimum solution and defuzzification by graded mean integration representation method. A numerical example is given to demonstrate the applicability of the purposed model and sensitivity analysis is carried out to reveal the impact of change in parameter values.
Any inventory system should focus on maintaining and increase levels of customer satisfaction while keeping inventory costs within predetermined time frames. The only way to increase the profit is to manage the product demand in accordance with market ups and downs and fluctuations. When it comes to low-life products that deteriorate quickly, such as milk products and vegetables. Additionally, it is not feasible to fix the demand for such things during the course of the products lifecycle. The stability of the production process, uncertainty regarding the magnitude of future requests, uncertainty regarding inventory costs, uncertainty regarding deterioration, etc., are some key factors that determine whether inventory control is successful or not.
In practice, parameters change over time, depending on the circumstances. In this research, a production inventory model for deteriorating products under the influence of inflation is considered. In today’s unstable economy, especially for long-term investments, the effects of inflation cannot be ignored because uncertainty regarding future inflation may also affect the ordering strategy. As inflation devalues currency, this effect of inflation should also be taken into account. Tayal et al. [
Some parameters have ambiguous definitions or not clearly defined; their values are approximated based on subjective beliefs. In order to evaluate the optimal solution for the model in various diverse circumstances, the inventory model is solved in a fuzzy environment. Shekarian et al. [
Roy et al. [
In this paper, an inventory model using a fuzzy approach was built to ascertain the ideal time period for the inventory cycle and the lowest possible total average costs. It is the production inventory model created for deteriorating products in which the production rate linearly dependent on the demand. There are four stages in the model. In the first two stages, with starting production rate and following change in production rate, inventory level rises. The third stage occurs when demand, which is based on selling price, is depreciating the inventory level. This stage occurs after inventory accumulates but before deterioration begins. Deterioration occurs in the fourth stage, which is two-parameter Weibull deterioration. Backorders are not taken into account. The optimum solution is determined using hexagonal fuzzy numbers, and the defuzzification process is handled using the graded mean integration representation approach.
Definition 2.1. [
A fuzzy set A ˜ on the given universal set is a set of order pairs A ˜ = { ( x , μ A ˜ ( x ) ) : x ∈ X } , where, μ A ˜ : X → [ 0 , 1 ] is a mapping called membership function. The membership function is also a degree of compatibility or a degree of truth of x in A ˜ .
Definition 2.2. [
The α-cut of A ˜ is defined by, A α = { x : μ A ˜ ( x ) = α , α ≥ 0 }
If R is a real line, then a fuzzy number is a fuzzy set A ˜ with membership function μ A ˜ : X → [ 0 , 1 ] , having following properties,
1) A ˜ is normal i.e., there exists x ∈ R such that μ A ˜ ( x ) = 1 ;
2) A ˜ is piecewise continuous;
3) sup p ( A ˜ ) = c l { x ∈ R : μ A ˜ ( x ) > 0 } ;
4) A ˜ is a convex fuzzy set.
Definition 2.3. [
The fuzzy number set A ˜ = ( a , b , c , d , e , f ) where, a ≤ b ≤ c ≤ d ≤ e ≤ f and defined on R, is called the Hexagonal fuzzy number, if the membership function of A ˜ is given by,
μ A ˜ ( x ) = { L 1 ( x ) = 1 2 ( x − a b − a ) , a ≤ x ≤ b L 2 ( x ) = 1 2 + 1 2 ( x − b c − d ) , b ≤ x ≤ c 1 , c ≤ x ≤ d R 1 ( x ) = 1 − 1 2 ( x − d e − d ) , d ≤ x ≤ e R ( x ) = 1 2 ( f − x f − e ) , e ≤ x ≤ f 0 , Otherwise
The α-cut of A ˜ = ( a , b , c , d , e , f ) , 0 ≤ α ≤ 1 is A ( α ) = [ A L ( α ) , A R ( α ) ] where,
A L 1 ( α ) = a + ( b − a ) α = L 1 − 1 ( α ) ,
A L 2 ( α ) = b + ( c − b ) α = L 2 − 1 ( α ) ,
A R 1 ( α ) = e + ( e − d ) α = R 1 − 1 ( α ) ,
A R 2 ( α ) = f + ( f − e ) α = R 2 − 1 ( α ) ,
And,
L − 1 ( α ) = L 1 − 1 ( α ) + L 2 − 1 ( α ) 2 = a + b + ( c − a ) α 2
R − 1 ( α ) = R 1 − 1 ( α ) + R 2 − 1 ( α ) 2 = e + f + ( d − f ) α 2
Definition 2.4. [
Suppose A ˜ = ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) and B ˜ = ( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 ) are two hexagonal fuzzy numbers, then arithmetical operations are defined as,
1) A ˜ ⊕ B ˜ = ( a 1 + b 1 , a 2 + b 2 , a 3 + b 3 , a 4 + b 4 , a 5 + b 5 , a 6 + b 6 )
2) A ˜ ⊗ B ˜ = ( a 1 b 1 , a 2 b 2 , a 3 b 3 , a 4 b 4 , a 5 b 5 , a 6 b 6 )
3) A ˜ − B ˜ = ( a 1 − b 1 , a 2 − b 2 , a 3 − b 3 , a 4 − b 4 , a 5 − b 5 , a 6 − b 6 )
4) A ˜ ⊘ B ˜ = ( a 1 b 1 , a 2 b 2 , a 3 b 3 , a 4 b 4 , a 5 b 5 , a 6 b 6 )
5) α ⊕ A ˜ = { ( α a 1 , α a 2 , α a 3 , α a 4 , α a 5 , α a 6 ) α ≥ 0 ( α a 6 , α a 5 , α a 4 , α a 3 , α a 2 , α a 1 ) α < 0
Definition 2.5. [
If A ˜ = ( a , b , c , d , e , f ) is a hexagonal fuzzy number, then the graded mean integration representation (GMIR) method of A ˜ is defined as,
P ( A ˜ ) = ∫ 0 W A h 2 ( L − 1 ( h ) + R − 1 ( h ) 2 ) d h ∫ 0 W A h d h , with 0 ≤ W A ≤ 1 .
P ( A ˜ ) = a + 3 b + 2 c + 2 d + 3 e + f 12
The following notations and assumptions are considered throughout the paper:
1) Inventory cycle for single product is considered.
2) The demand rateD(p) is dependent on selling price p i.e. D(p) = ηp−γ where, η > 0, γ > 0, where, η is scaling factor, γ is index of price elasticity.
3) The production rate is linearly dependent on demand, that is, P(p) = λ·D(p) where, λ > 1 and production rate is greater than demand rateD(p).
4) Lead time is considered to be negligible.
5) The inflationary effects and time value of money are taken into consideration.
6) The setup cost for inventory is constant.
7) No Shortages are allowed.
8) The deterioration of the products starts after a certain fix time. The rate of deterioration at time (td, T) is θ(t) = αβtβ−1 which is two parameter Weibull distribution where, α represents scale parameter and β represents shape parameter. There is no deterioration before time td.
As shown in
inventory levels start rising at a rate a(λ − 1)D(p). When the inventory level becomes Q2 at timet = t2, production stopped. The inventory level is depleted due to demand alone during the time interval (t2, td) and at t= td inventory level becomes Q3. In the time interval (td, T), inventory level starts decreasing due to both deterioration as well as demand rate and then reaches to zero level at time t = T.
The differential equations representing the inventory model are,
d I 1 ( t ) d t = ( λ − 1 ) D ( p ) , 0 ≤ t ≤ t 1 (1)
d I 2 ( t ) d t = a ( λ − 1 ) D ( p ) , t 1 ≤ t ≤ t 2 (2)
d I 3 ( t ) d t = − D ( p ) , t 2 ≤ t ≤ t d (3)
d I 4 ( t ) d t + α β t β − 1 I 4 ( t ) = − D ( p ) , t d ≤ t ≤ T (4)
The boundary conditions are,
I 1 ( 0 ) = 0 , I 1 ( t 1 ) = I 2 ( t 1 ) = Q 1 , I 2 ( t 2 ) = I 3 ( t 2 ) = Q 2 , I 3 ( t d ) = I 4 ( t d ) = Q 3 , I 1 ( T ) = 0 (5)
The solutions of differential equations above are given by,
I 1 ( t ) = η ( λ − 1 ) t p γ (6)
I 2 ( t ) = η a ( λ − 1 ) t p γ + η ( 1 − a ) ( λ − 1 ) t 1 p γ (7)
I 3 ( t ) = η p γ ( t d − t ) + η p γ [ ( T − t d ) + α β + 1 ( T β + 1 − t d β + 1 ) ] e − α t d β (8)
I 4 ( t ) = η p γ [ ( T − t ) + α β + 1 ( T β + 1 − t β + 1 ) ] e − α t β (9)
Using initial boundary conditions,
Q 1 = η ( λ − 1 ) t 1 p γ (10)
Q 2 = η ( λ − 1 ) t 1 p γ + η a ( λ − 1 ) ( t 2 − t 1 ) p γ
Q 3 = η p γ [ ( T − t d ) + α β + 1 ( T β + 1 − t d β + 1 ) ] e − α t d β (11)
The different costs included in total cost are as follows, considering the influence of inflation and time value of money.
Total cost of inventory cycle per unit time is,
T C = 1 T [ ProductionCost + HoldingCost + DeteriorationCost + SetupCost ]
i) ProductionCost = C P C [ ∫ 0 t 1 η λ p γ e − ( r − i ) t d t + ∫ t 1 t 2 a η λ p γ e − ( r − i ) t d t ] = C P C η λ p γ [ a ( t 2 − ( r − i ) t 2 2 2 ) + ( 1 − a ) ( t 1 − ( r − i ) t 1 2 2 ) ] (12)
ii) HoldingCost = C H C [ ∫ 0 t 1 I 1 ( t ) e − ( r − i ) t d t + ∫ t 1 t 2 I 2 ( t ) e − ( r − i ) t d t + ∫ t 2 t d I 3 ( t ) e − ( r − i ) t d t + ∫ t d T I 4 ( t ) e − ( r − i ) t d t ]
= C H C { η a ( λ − 1 ) p γ ( t 2 2 2 − ( r − i ) t 2 3 3 ) + η ( r − i ) p γ ( t d 3 − t 2 3 3 − t d ( t d 2 − t 2 2 ) 2 ) + η ( λ − 1 ) ( 1 − a ) p γ ( t 1 t 2 − t 1 2 2 + ( r − i ) t 1 3 6 − ( r − i ) t 1 t 2 2 2 ) + η p γ ( t d − t 2 − r − i 2 ( t d 2 − t 2 2 ) ) [ ( T − t d ) + α β + 1 ( T β + 1 − t d β + 1 ) ] e − α t d β + η ( t d − t 2 ) 2 2 p γ + η p γ [ ( T − t d ) 2 2 + α β ( T β + 2 − t d β + 2 ) ( β + 1 ) ( β + 2 ) − α T t d ( T β − t d β ) β + 1 − ( r − i ) ( T 3 6 − T t d 2 2 + t d 3 3 ) − α ( r − i ) T β + 3 2 ( β + 3 ) + α ( r − i ) T β + 1 t d 2 2 ( β + 1 ) − α ( r − i ) t 2 β + 3 ( β + 1 ) ( β + 3 ) ] } (13)
iii) DeteriorationCost = C D C [ ∫ t d T θ ( t ) I 4 ( t ) e − ( r - i ) t d t ] = C D C { η α β p γ [ T β + 1 β ( β + 1 ) − T t d β β + t d β + 1 β + 1 + ( r − i ) ( T β + 2 − t d β + 2 ) β + 2
+ ( r − i ) T ( T β + 1 − t d β + 1 ) β + 1 ] } (14)
iv) Setup Cost = A (15)
Then, The Total cost per unit time for inventory cycle is,
T C ( T ) = 1 T [ C P C η λ p γ [ a ( t 2 − ( r − i ) t 2 2 2 ) + ( 1 − a ) ( t 1 − ( r − i ) t 1 2 2 ) ] + C H C { η a ( λ − 1 ) p γ ( t 2 2 2 − ( r − i ) t 2 3 3 ) + η ( λ − 1 ) ( 1 − a ) p γ ( t 1 t 2 − t 1 2 2 + ( r − i ) t 1 3 6 − ( r − i ) t 1 t 2 2 2 ) + η ( r − i ) p γ ( t d 3 − t 2 3 3 − t d ( t d 2 − t 2 2 ) 2 ) + η ( t d − t 2 ) 2 2 p γ + η p γ ( t d − t 2 − r − i 2 ( t d 2 − t 2 2 ) ) [ ( T − t d ) + α β + 1 ( T β + 1
− t d β + 1 ) ] e − α t d β + η p γ [ ( T − t d ) 2 2 + α β ( T β + 2 − t d β + 2 ) ( β + 1 ) ( β + 2 ) − α T t d ( T β − t d β ) ( β + 1 ) − ( r − i ) ( T 3 6 − T t d 2 2 + t d 3 3 ) − α ( r − i ) T β + 3 2 ( β + 3 ) + α ( r − i ) T β + 1 t d 2 2 ( β + 1 ) − α ( r − i ) t 2 β + 3 ( β + 1 ) ( β + 3 ) ] } + C D C { η α β p γ [ T β + 1 β ( β + 1 ) − T t d β β + t d β + 1 β + 1 + ( r − i ) ( T β + 2 − t d β + 2 ) β + 2 + ( r − i ) T ( T β + 1 − t d β + 1 ) β + 1 ] } + A ] (16)
Let t 1 = c 1 T , t 2 = c 2 T , t d = c 3 T such that, 0 < c 1 , c 2 , c 3 < 1 and T > t d > t 2 > t 1
T C ( T ) = 1 T [ C P C η λ p γ [ a ( c 2 T − ( r − i ) c 2 2 T 2 2 ) + ( 1 − a ) ( c 1 T − ( r − i ) c 1 2 T 2 2 ) ] + C H C { η a ( λ − 1 ) p γ ( c 2 2 T 2 2 − ( r − i ) c 2 3 T 3 3 ) + η ( λ − 1 ) ( 1 − a ) p γ ( c 1 c 2 T 2 − c 1 2 T 2 2 + ( r − i ) c 1 3 T 3 6 − ( r − i ) c 1 T ( c 2 2 T 2 ) 2 ) + η ( r − i ) p γ ( c 3 3 T 3 − c 2 3 T 3 3 − t d ( c 3 2 T 2 − c 2 2 T 2 ) 2 ) + η ( c 3 T − c 2 T ) 2 2 p γ + η p γ ( c 3 T – c 2 T − r − i 2 ( c 3 2 T 2 − c 2 2 T 2 ) ) [ ( T − c 3 T ) + α β + 1 ( T β + 1
− c 3 β + 1 T β + 1 ) ] e − α c 3 β T β + η p γ [ ( T − c 3 T ) 2 2 + α β ( T β + 2 − c 3 β + 2 T β + 2 ) ( β + 1 ) ( β + 2 )
− α c 3 T 2 ( T β − c 3 β T β ) β + 1 − ( r − i ) ( T 3 6 − T ( c 3 2 T 2 ) 2 + c 3 3 T 3 3 ) − α ( r − i ) T β + 3 2 ( β + 3 ) + α ( r − i ) T β + 1 ( c 3 2 T 2 ) 2 ( β + 1 ) − α ( r − i ) c 2 β + 3 T β + 3 ( β + 1 ) ( β + 3 ) ] } + C D C { η α β p γ [ T β + 1 β ( β + 1 ) − c 3 β T β + 1 β + c 3 β + 1 T β + 1 β + 1 + ( r − i ) ( T β + 2 − c 3 β + 2 T β + 2 ) β + 2 + ( r − i ) T ( T β + 1 − c 3 β + 1 T β + 1 ) β + 1 ] } + A ] (17)
By minimizing the total cost TC(T), the following differential equation can be solved to determine the optimum value of T.
d T C ( T ) d T = 0 satisfying the condition, d 2 T C ( T ) d T 2 > 0
Fuzzy Model
Due to uncertainty in the market, all parameters cannot be defined precisely, hence considering A ˜ , C ˜ H C , C ˜ D C , C ˜ P C may change within some limits.
Let A ˜ = ( A 1 , A 2 , A 3 , A 4 , A 5 , A 6 ) , C ˜ H C = ( C H C 1 , C H C 2 , C H C 3 , C H C 4 , C H C 5 , C H C 6 ) , C ˜ D C = ( C D C 1 , C D C 2 , C D C 3 , C D C 4 , C D C 5 , C D C 6 ) , C ˜ P C = ( C P C 1 , C P C 2 , C P C 3 , C P C 4 , C P C 5 , C P C 6 ) , are Hexagonal fuzzy numbers.
In a fuzzy sense, the total cost of the model per unit of time is given by,
T C ˜ ( T ) = 1 T [ C ˜ P C η λ p γ [ a ( c 2 T − ( r − i ) c 2 2 T 2 2 ) + ( 1 − a ) ( c 1 T − ( r − i ) c 1 2 T 2 2 ) ] + C ˜ H C { η a ( λ − 1 ) p γ ( c 2 2 T 2 2 − ( r − i ) c 2 3 T 3 3 ) + η ( λ − 1 ) ( 1 − a ) p γ ( c 1 c 2 T 2 − c 1 2 T 2 2 + ( r − i ) c 1 3 T 3 6 − ( r − i ) c 1 T ( c 2 2 T 2 ) 2 ) + η ( r − i ) p γ ( c 3 3 T 3 − c 2 3 T 3 3 − t d ( c 3 2 T 2 − c 2 2 T 2 ) 2 ) + η ( c 3 T − c 2 T ) 2 2 p γ + η p γ ( c 3 T – c 2 T − r − i 2 ( c 3 2 T 2 − c 2 2 T 2 ) ) [ ( T − c 3 T ) + α β + 1 ( T β + 1
− c 3 β + 1 T β + 1 ) ] e − α c 3 β T β + η p γ [ ( T − c 3 T ) 2 2 + α β ( T β + 2 − c 3 β + 2 T β + 2 ) ( β + 1 ) ( β + 2 ) − α c 3 T 2 ( T β − c 3 β T β ) β + 1 − ( r − i ) ( T 3 6 − T ( c 3 2 T 2 ) 2 + c 3 3 T 3 3 ) − α ( r − i ) T β + 3 2 ( β + 3 ) + α ( r − i ) T β + 1 ( c 3 2 T 2 ) 2 ( β + 1 ) − α ( r − i ) c 2 β + 3 T β + 3 ( β + 1 ) ( β + 3 ) ] }
+ C ˜ D C { η α β p γ [ T β + 1 β ( β + 1 ) − c 3 β T β + 1 β + c 3 β + 1 T β + 1 β + 1 + ( r − i ) ( T β + 2 − c 3 β + 2 T β + 2 ) β + 2 + ( r − i ) T ( T β + 1 − c 3 β + 1 T β + 1 ) β + 1 ] } + A ˜ ] (18)
Let T C ˜ i ( T ) be the corresponding total cost obtained by replacing A ˜ i , C ˜ H C i , C ˜ D C i , C ˜ P C i in Equation (17) for i = 1 , 2 , 3 , 4 , 5 , 6 . Using graded mean representation to defuzzify the fuzzy total cost T C ˜ ( T ) .
We get,
T C ˜ ( T ) = 1 12 [ T C ˜ 1 ( T ) + 2 T C ˜ 2 ( T ) + 3 T C ˜ 3 ( T ) + 3 T C ˜ 4 ( T ) + 2 T C ˜ 5 ( T ) + T C ˜ 6 ( T ) ]
By minimizing the total cost T C ˜ ( T ) , the following differential equation can be solved to determine the optimum value of T.
d T C ˜ ( T ) d T = 0 satisfying the condition, d 2 T C ˜ ( T ) d T 2 > 0
The Economic Production Quantity (EPQ) for inventory model with inventory cycle length (T) can obtained as,
EPQ* = Total demand during production period + total demand after production stopped + total demand during deterioration + total number of deteriorated items
Q ∗ = η ( λ − 1 ) t 1 p γ + η a ( λ − 1 ) ( t 2 − t 1 ) p γ + η ( λ − 1 ) ( t d − t 2 ) p γ + η p γ [ ( T − t d ) + α β + 1 ( T β + 1 − t d β + 1 ) − α t d β ( T − t d ) ] (19)
Consider following parametric values.
CPC = Rs 10/unit, CDC = Rs 12/unit, CHC = Rs 7/unit, A = Rs 4000/order, α = 0.01, β =2, p = 20, γ = 2.1, λ = 4, a = 1.5, η = 20,000, c1 = 0.3, c2 = 0.5, c3 = 0.7, r = 0.5, i = 1.2.
The solution of crisp model is
T = 2.7220, TC(T) = 3507.74, t1 = 0.8166, t2 = 1.3610, td = 1.9054, Q* = 232.5356
C ˜ P C = (7, 8, 9, 11, 12, 13), C ˜ H C = (4, 5, 6, 8, 9, 10), C ˜ D C = (9, 10, 11, 13, 14, 15), A ˜ = (1000, 2000, 3500, 4500, 5000, 6000), α = 0.01, β =2, p = 20, γ = 2.1, λ = 4, a = 1.5, η = 20,000, c1 = 0.3, c2 = 0.5, c3 = 0.7, r = 0.5, i = 1.2.
The solution of fuzzy model is given by,
T = 2.6270, T C ˜ ( T ) = 3383.11, t1 = 0.7881, t2 = 1.3135, td = 1.8390, Q* = 224.38
The Time parameters of inventory cycle for crisp and fuzzy model are compared in
Taking into account the above numerical example of the fuzzy model for sensitivity analysis to examine the impact of changing various inventory model parameters.
As shown in
| T | TC | Q | t1 | t2 | td | Q1 | Q2 | Q3 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| A | −20% | 2.4591 | 3419.97 | 251.98 | 0.7377 | 1.2296 | 1.7214 | 98.42 | 196.83 | 33.25 |
| −10% | 2.5824 | 3578.64 | 264.67 | 0.7747 | 1.2912 | 1.8077 | 103.35 | 206.7 | 34.97 | |
| 0% | 2.6972 | 3730.15 | 276.49 | 0.8092 | 1.3486 | 1.888 | 107.95 | 215.89 | 36.57 | |
| 10% | 2.8049 | 3875.53 | 287.58 | 0.8415 | 1.4025 | 1.9634 | 112.26 | 224.51 | 38.07 | |
| 20% | 2.9065 | 4015.59 | 298.05 | 0.8719 | 1.4533 | 2.0346 | 116.32 | 232.64 | 39.5 | |
| CPC | −20% | 2.6695 | 3304.28 | 273.64 | 0.8009 | 1.3348 | 1.8687 | 106.84 | 213.67 | 36.18 |
| −10% | 2.6353 | 3454.6 | 270.12 | 0.7906 | 1.3177 | 1.8447 | 105.47 | 210.94 | 35.7 | |
| 0% | 2.6021 | 3524.01 | 266.7 | 0.7806 | 1.3011 | 1.8215 | 104.14 | 208.28 | 35.24 | |
| 10% | 2.5699 | 3753.58 | 263.38 | 0.771 | 1.285 | 1.7989 | 102.85 | 205.7 | 34.79 | |
| 20% | 2.5387 | 3902.29 | 260.17 | 0.7616 | 1.2694 | 1.7771 | 101.6 | 203.2 | 34.36 | |
| CDC | −20% | 2.6043 | 3603.22 | 266.92 | 0.7813 | 1.3022 | 1.823 | 104.23 | 208.45 | 35.27 |
| −10% | 2.6032 | 3603.79 | 266.81 | 0.7809 | 1.3016 | 1.8222 | 104.18 | 208.37 | 35.26 | |
| 0% | 2.6021 | 3604.36 | 266.7 | 0.7806 | 1.3011 | 1.8215 | 104.14 | 208.28 | 35.24 | |
| 10% | 2.601 | 3604.92 | 266.58 | 0.7803 | 1.3005 | 1.8207 | 104.09 | 208.19 | 35.22 | |
| 20% | 2.5999 | 3605.49 | 266.47 | 0.7799 | 1.2999 | 1.8199 | 104.05 | 208.1 | 35.21 | |
| CHC | −20% | 2.7672 | 3459.45 | 283.7 | 0.8302 | 1.3836 | 1.937 | 110.75 | 221.49 | 37.54 |
| −10% | 2.68 | 3533.46 | 274.72 | 0.804 | 1.34 | 1.876 | 107.26 | 214.51 | 36.33 | |
| 0% | 2.6021 | 3604.36 | 266.7 | 0.7806 | 1.3011 | 1.8215 | 104.14 | 208.28 | 35.24 | |
| 10% | 2.5318 | 3672.51 | 259.46 | 0.7595 | 1.2659 | 1.7723 | 101.33 | 202.65 | 34.26 | |
| 20% | 2.4679 | 3738.2 | 252.89 | 0.7404 | 1.2339 | 1.7275 | 98.77 | 197.54 | 33.38 | |
| η | −20% | 2.8764 | 2972.88 | 196.63 | 0.8629 | 1.4382 | 2.0135 | 76.74 | 153.49 | 26.05 |
| −10% | 2.7423 | 3181.36 | 210.85 | 0.8227 | 1.3712 | 1.9196 | 82.21 | 164.63 | 27.9 | |
| 0% | 2.627 | 3383.11 | 224.38 | 0.7881 | 1.3135 | 1.8389 | 87.61 | 175.23 | 29.66 | |
| 10% | 2.5264 | 3579.14 | 237.33 | 0.7579 | 1.2632 | 1.7685 | 92.68 | 185.37 | 31.34 | |
| 20% | 2.4375 | 3770.22 | 249.76 | 0.7313 | 1.2188 | 1.7063 | 97.55 | 196.1 | 32.95 | |
| λ | −20% | 2.8506 | 2985.05 | 192.87 | 0.8551 | 1.4253 | 1.9954 | 69.72 | 139.44 | 32.26 |
| −10% | 2.7317 | 3186.84 | 209.07 | 0.8195 | 1.3659 | 1.9122 | 78.96 | 157.92 | 30.87 | |
| 0% | 2.627 | 3383.11 | 224.38 | 0.7881 | 1.3135 | 1.8389 | 87.61 | 175.23 | 29.66 | |
| 10% | 2.5341 | 3574.59 | 238.95 | 0.7602 | 1.2671 | 1.7739 | 95.78 | 191.57 | 28.58 | |
| 20% | 2.4507 | 3761.83 | 252.86 | 0.7352 | 1.2254 | 1.7155 | 103.53 | 207.06 | 27.61 |
1) An increase in set-up cost A, increases the total average cost TC(T), production time (t1 and t2), non-production time (t3), optimum inventory cycle time (T) and economic production quantity (Q), Maximum level of inventory (Q1, Q2 and Q3) also increases.
2) An increase in the purchase cost (CPC), decreses economic production quantity (Q), optimum inventory cycle time (T), production time (t1 and t2), non-production time (t3) and Maximum level of inventory (Q1,Q2 and Q3), but the total average cost TC(T) increases.
3) An increase in the Deteriorating cost (CDC), decreses economic production quantity (Q), optimum inventory cycle time (T) production time (t1 and t2), non-production time (t3) and Maximum level of inventory (Q1,Q2 and Q3), but the total average cost TC(T) increases.
4) With the increase in holding cost (CHC), it is observed that, economic production quantity (Q), optimum inventory cycle time (T) production time (t1 and t2), non-production time (t3) and Maximum level of inventory (Q1,Q2 andQ3) decreses, but the total average cost TC(T) increases.
5) An increase in demand coefficient η, increases the total average cost TC(T), economic production quantity (Q) and Maximum level of inventory (Q1,Q2 and Q3) but optimum inventory cycle time (T), production time (t1 andt2), non-production time (t3) decreses.
6) With increase in the value of λ, the total average cost TC(T), economic production quantity (Q) and Maximum level of inventory (Q1,Q2 and Q3) increases but optimum inventory cycle time (T), production time (t1 and t2), non-production time (t3) decreses.
In the developed production inventory model, inflation and time value of money under fuzzy environment is considered, where demand is a function of selling price. Production rate, demand rate and deterioration rate are the three important factors in the inventory model, whereas production rate is considered to be dependent on demand rate, and inventory level increases with two production rates at two stages of inventory cycle. As the production stops, the inventory level diminishes only due to demand, before the deterioration period starts. In the last stage, inventory reaches zero level due to demand and deterioration rate which is following two parameter Weibull distribution. As the inventory level reaches zero production is started again instantly. Shortages are not considered in this model.
The optimum solution for total average cost, economic production quantity and Maximum level of inventory (Q1, Q2 and Q3), inventory cycle time (T), production time (t1 and t2), non-production time (t3) is obtained for crisp model as well as fuzzy model. Hexagonal fuzzy numbers and for defuzzification graded mean integration representation method are used for the fuzzy model. The back-order is not considered. Hexagonal fuzzy numbers are used to derive optimum solution and defuzzification by graded mean integration representation method. A numerical example is given to demonstrate the applicability of the purposed model. By comparing the results of crisp model and fuzzy model, it can be concluded that, Fuzzy model is more beneficial.
In future aspect, one can develop this paper by adding shortages with fully backlogging or with partial backlogging.
The authors are thankful to the anonymous reviewers for their thoughtful comments and suggestions that helped throughout the submission process. This research work received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
The authors declare no conflicts of interest regarding the publication of this paper.
Shaikh, T.S. and Gite, S.P. (2022) Fuzzy Inventory Model with Variable Production and Selling Price Dependent Demand under Inflation for Deteriorating Items. American Journal of Operations Research, 12, 233-249. https://doi.org/10.4236/ajor.2022.126013