Inventories are necessary to keep the commodities in balance. Controlling and keeping track of tangible goods supplies is a difficulty that every business, regardless of industry, faces. There are a number of reasons why companies ought to maintain inventory. It is not possible for goods to reach specific systems precisely when they are needed, for either economic or physical reasons. A deficiency in inventory management can also impede the production process and ultimately raise the cost per unit of production. Inventory management is critical because it enables the company to successfully handle two key challenges: keeping enough inventory on hand to facilitate seamless production and sales processes and reducing inventory expenditures to boost profitability.

Inventory control is an essential part which is to be taken care of for smooth and efficient facilities and an increase in profit. The commodities that undergo deterioration as time passes are the common challenge for managing inventories. Some items do not start deteriorating instantaneously like Fruits, Milk, Vegetables, Meat, Medicines, etc. but after a certain period of time, the deterioration starts speedily.

The first inventory model for deteriorating items was developed by Whitin [1] . Exponentially decaying inventory model was studied by Ghare & Scharder [2] . Covert & Philip [3] considered Weibull deterioration rate in inventory model. Datta & Pal [4] studied the order level inventory model with power demand and variable deterioration rate. Giri & Chaudhuri [5] developed a deterministic model for deteriorating items where demand is stock dependent. In the last two decades, economic conditions changed tremendously. Thus, the time value of money cannot be ignored. Ouyang et al. [6] studied the inventory model for deteriorating items under the conditions of the time value of money and inflation. Mishra [7] developed an inventory model with a controllable deteriorating rate and time dependent demand. Sharma et al. [8] studied the inventory model with stock dependent demand under inflation. Zhao [9] considered Trapezoidal type demand with Weibull distribution deterioration and partial backlogging. Uthayakumar & Karuppasamy [10] introduced an inventory model in healthcare industries with different types of time dependent demand for deteriorating items.

In the theoretical inventory model, the parameters are certain. But, in real life situations, these parameters may not follow any certainty. In such cases, they are treated as fuzzy parameters. Fuzzy set theory was first introduced by Zadeh [11] in 1965. Fuzzy set theory is highly applicable to inventory models involving marketing parameters. Resulting in a large number of researches published using fuzzy approach in inventory control as well as other fields. Shekarian et al. [12] developed a literature review of the fuzzy inventory model which identified and classified common characteristics of these models.

K. Jaggi et al. [13] , and Kumar & Rajput [14] developed a fuzzy inventory model for deteriorating items with time varying demand. Mandal & Islam [15] investigated a fuzzy EOQ model with constant demand and fully backlogged shortages. Mohanty & Tripathy [16] studied an inventory model with exponentially decreasing demand and fuzzified costs. Sahoo et al. [17] analyzed three rates of fuzzy inventory model for deteriorating items with shortages. Biswas & Islam [18] developed a production inventory model where demand is dependent on selling price and advertisement. Indrajitsingha et al. [19] analyzed inventory model for non-instantaneous deteriorating items of selling price dependent demand during the pandemic Covid-19, where the deteriorating rate is considered to be time dependent.

As seasons change, we always encounter variations in the price of commodities. Therefore, the concept of selling price dependent demand is considered, which indicates the tendency of the change in demand for certain deteriorating items to the change in the selling price. Also, the production process time period may be impacted in certain circumstances by unforeseen and unanticipated events, allowing for the practical achievement of optimality. Consequently, when executing various industrial tasks, the ideal answer is typically not precisely identified. As such, it is quite challenging for decision makers to give a precise figure that would adequately capture the likely and necessary characteristics of production inventory difficulties. Fuzzy numbers enable to get through this challenge.

In this paper, a fuzzy inventory model with selling price dependent demand is considered. The shortages are allowed and fully backlogged. The deteriorating items maintain their quality for a certain period of time, so there is no deterioration initially and then deterioration occurs which follows two-parameter Weibull distribution. For a fuzzy model, parameters like demand, holding cost, deterioration cost, ordering cost, purchase cost, and shortage cost are assumed to be Hexagonal fuzzy numbers. The Graded mean integration representation method is used for defuzzification.

The following notations and assumptions are considered to develop the inventory model:

Let q₁ be the total quantity at the beginning of each cycle and after fulfilling q₂ units of backorder inventory. The described inventory model is of deteriorating items which starts deteriorating after a certain period of time. Let T be the length of the cycle. In the time interval [0, t_m], there is no deterioration at all. The deterioration starts at t = t_m. During the interval [t_m, t₁], inventory level decreases due to the demand as well as deterioration. At t = t₁, inventory falls to zero. The time interval [t₁, T] is shortage period, which is fully backlogged. Let I₁ (t), I₂ (t), I₃ (t) be the inventory levels at any time t, in the interval [0, t_m], [t_m, t₁] and [t₁, T] respectively. The model is represented in Figure 1.

The differential equations for the inventory model are given as,

d I 1 ( t ) d t = − η s β 0 ≤ t ≤ t m (1)

d I 2 ( t ) d t + θ ( t ) ⋅ I ( t ) = − η s β (2)

d I 3 ( t ) d t = − η s β t 1 ≤ t ≤ T (3)

where, θ ( t ) = γ λ t λ − 1 , 0 ≤ γ ≤ 1 , λ ≥ 1 .

The boundary conditions are,

I ( 0 ) = q 1 , I ( t 1 ) = 0 and I 1 ( t m ) = I 2 ( t m ) (4)

The solution of Equations (1), (2) and (3) is given by,

I 1 ( t ) = η s β ( t m − t ) + η s β [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ (5)

I 2 ( t ) = η s β [ ( t 1 − t ) + γ λ + 1 ( t 1 λ + 1 − t λ + 1 ) ] e − γ t λ (6)

I 3 ( t ) = η s β ( t 1 − t ) (7)

Also, using initial boundary condition, I (0) = q₁,

We get,

q 1 = η s β t m + η s β [ ( t 1 – t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ (8)

The total ordering quantity Q is the sum of on-hand inventory and back-order inventory which is,

Q = q 1 + q 2

q 2 = ∫ t 1 T I 3 ( t ) d t = − η 2 s β ( T − t 1 ) 2 (9)

Total inventory cost per unit time for the model during a cycle is given by,

C ( t 1 , T ) = 1 T [ PurchaseCost + HoldingCost + DeteriorationCost   + ShortageCost + OrderingCost ]

Now,

1)

PurchaseCost = C P C ⋅ Q = C P C ⋅ ( q 1 + q 2 )                                         = C P C ⋅ { η s β t m + η s β [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η 2 s β ( T − t 1 ) 2 } (10)

2)

HoldingCost = C H C ⋅ [ ∫ 0 t m I 1 ( t ) d t + ∫ t m t 1 I 2 ( t ) d t ]                                     = C H C ⋅ { η 2 s β t m 2 + η t m s β [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ                                           + η s β [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] } (11)

3)

DeteriorationCost = C D C ⋅ [ ∫ t m t 1 θ ( t ) ⋅ I 2 ( t ) d t ]                                                   = C D C ⋅ { η γ λ s β [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } (12)

4)

ShortageCost = − C S C [ ∫ t 1 T I 3 ( t ) d t ] = C S C [ η 2 s β ( T − t 1 ) 2 ] (13)

5)

Ordering Cost = C_OC (14)

Hence, Total inventory cost per unit time is,

C ( t 1 , T ) = 1 T [ C P C ⋅ { η s β t m + η s β [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η 2 s β ( T − t 1 ) 2 }       + C H C ⋅ { η 2 s β t m 2 + η t m s β [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ       + η s β [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] }       + C D C ⋅ { η γ λ s β [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C S C [ η 2 s β ( T − t 1 ) 2 ] + C O C ] (15)

For minimization of the total cost C (t₁, T), the optimal value of t₁ and T can be obtained by solving the following differential equation,

∂ C ( t 1 , T ) ∂ t 1 = 0 and ∂ C ( t 1 , T ) ∂ T = 0 ,

And it should satisfy the condition

( ∂ 2 C ( t 1 , T ) ∂ t 1 2 ) ( ∂ 2 C ( t 1 , T ) ∂ T 2 ) − ( ∂ 2 C ( t 1 , T ) ∂ t 1 ⋅ ∂ T ) > 0 .

Fuzzy Model

Due to uncertainty in the market, it is not easy to define all parameters precisely, we assume some of these parameters η ˜ , β ˜ , C ˜ H C , C ˜ D C , C ˜ O C , C ˜ P C , C ˜ S C may change within some limits.

Let η ˜ = ( η 1 , η 2 , η 3 , η 4 , η 5 , η 6 ) , β ˜ = ( β 1 , β 2 , β 3 , β 4 , β 5 , β 6 ) , C ˜ S C = ( C S C 1 , C S C 2 , C S C 3 , C S C 4 , C S C 5 , C S C 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 ˜ O C = ( C O C 1 , C O C 2 , C O C 3 , C O C 4 , C O C 5 , C O 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.

The corresponding total inventory cost in fuzzy environment is given by,

C ˜ ( t 1 , T ) = 1 T [ C ˜ P C ⋅ { η ˜ s β ˜ t m + η ˜ s β ˜ [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 2 s β ˜ ( T − t 1 ) 2 }       + C ˜ H C ⋅ { η ˜ 2 s β ˜ t m 2 + η ˜ t m s β ˜ [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ       + η ˜ s β ˜ [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] }       + C ˜ D C ⋅ { η ˜ γ λ s β ˜ [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C ⋅ [ η ˜ 2 s β ˜ ( T − t 1 ) 2 ] + C ˜ O C ] (16)

Let C ˜ i ( t 1 , T ) be the corresponding total inventory cost obtained by replacing η ˜ i , β ˜ i , C ˜ H C i , C ˜ D C i , C ˜ O C i , C ˜ P C i , C ˜ S C i in Equation (16) for i = 1, 2, 3, 4, 5, 6.

The defuzzification of the fuzzy total cost C ˜ ( t 1 , T ) by graded mean representation is given by,

G C ˜ ( t 1 , T ) = 1 12 [ C ˜ 1 ( t 1 , T ) + 2 C ˜ 2 ( t 1 , T ) + 3 C ˜ 3 ( t 1 , T ) + 3 C ˜ 4 ( t 1 , T )     + 2 C ˜ 5 ( t 1 , T ) + C ˜ 6 ( t 1 , T ) ]

G C ˜ ( t 1 , T ) = 1 12 T [ C ˜ P C 1 ⋅ { η ˜ 1 s β ˜ 1 t m + η ˜ 1 s β ˜ 1 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 1 2 s β ˜ 1 ( T − t 1 ) 2 }       + C ˜ H C 1 ⋅ { η ˜ 1 2 s β ˜ 1 t m 2 + η ˜ 1 t m s β ˜ 1 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ       + η ˜ 1 s β ˜ 1 [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] }       + C ˜ D C 1 ⋅ { η ˜ 1 γ λ s β ˜ 1 [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C 1 ⋅ [ η ˜ 1 2 s β ˜ 1 ( T − t 1 ) 2 ] + C ˜ O C 1 ]

+   2 12 T [ C ˜ P C 2 ⋅ { η ˜ 2 s β ˜ 2 t m + η ˜ 2 s β ˜ 2 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 2 2 s β ˜ 2 ( T − t 1 ) 2 } +   C ˜ H C 2 ⋅ { η ˜ 2 2 s β ˜ 2 t m 2 + η ˜ 2 t m s β ˜ 2 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ +   η ˜ s β ˜ [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] } +   C ˜ D C 2 ⋅ { η ˜ 2 γ λ s β ˜ 2 [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C 2 ⋅ [ η ˜ 2 2 s β ˜ 2 ( T − t 1 ) 2 ] + C ˜ O C 2 ]

+   3 12 T [ C ˜ P C 3 ⋅ { η ˜ 3 s β ˜ 3 t m + η ˜ 3 s β ˜ 3 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 3 2 s β ˜ 3 ( T − t 1 ) 2 } +   C ˜ H C 3 ⋅ { η ˜ 3 2 s β ˜ 3 t m 2 + η ˜ 3 t m s β ˜ 3 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ +   η ˜ 3 s β ˜ 3 [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] } +   C ˜ D C 3 ⋅ { η ˜ 3 γ λ s β ˜ 3 [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C 3 ⋅ [ η ˜ 3 2 s β ˜ 3 ( T − t 1 ) 2 ] + C ˜ O C 3 ]

+   3 12 T [ C ˜ P C 4 ⋅ { η ˜ 4 s β ˜ 4 t m + η ˜ 4 s β ˜ 4 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 4 2 s β ˜ 4 ( T − t 1 ) 2 } +   C ˜ H C 4 ⋅ { η ˜ 4 2 s β ˜ 4 t m 2 + η ˜ 4 t m s β ˜ 4 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ +   η ˜ 4 s β ˜ 4 [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] } +   C ˜ D C 4 ⋅ { η ˜ 4 γ λ s β ˜ 4 [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C 4 ⋅ [ η ˜ 4 2 s β ˜ 4 ( T − t 1 ) 2 ] + C ˜ O C 4 ] +   2 12 T [ C ˜ P C 5 ⋅ { η ˜ 5 s β ˜ 5 t m + η ˜ 5 s β ˜ 5 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 5 2 s β ˜ 5 ( T − t 1 ) 2 } +   C ˜ H C 5 ⋅ { η ˜ 5 2 s β ˜ 5 t m 2 + η ˜ 5 t m s β ˜ 5 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ +   η ˜ 5 s β ˜ 5 [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] } +   C ˜ D C 5 ⋅ { η ˜ 5 γ λ s β ˜ 5 [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C ⋅ [ η ˜ 5 2 s β ˜ 5 ( T − t 1 ) 2 ] + C ˜ O C 5 ]

+   1 12 T [ C ˜ P C 6 ⋅ { η ˜ 6 s β ˜ 6 t m + η ˜ 6 s β ˜ 6 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ − η ˜ 6 2 s β ˜ 6 ( T − t 1 ) 2 } +   C ˜ H C 6 ⋅ { η ˜ 6 2 s β ˜ 6 t m 2 + η ˜ 6 t m s β ˜ 6 [ ( t 1 − t m ) + γ λ + 1 ( t 1 λ + 1 − t m λ + 1 ) ] e − γ t m λ +   η ˜ 6 s β ˜ 6 [ ( t 1 − t m ) 2 2 + γ λ ( t 1 λ + 2 − t m λ + 2 ) ( λ + 1 ) ( λ + 2 ) − γ t 1 t m ( t 1 λ − t m λ ) λ + 1 ] } +   C ˜ D C 6 ⋅ { η ˜ 6 γ λ s β ˜ 6 [ t 1 λ + 1 λ ( λ + 1 ) − t 1 t m λ λ + t m λ + 1 λ + 1 ] } + C ˜ S C 6 ⋅ [ η ˜ 6 2 s β ˜ 6 ( T − t 1 ) 2 ] + C ˜ O C 6 ]

For minimization of the total cost G C ˜ ( t 1 , T ) , the optimal value of t₁ and T can be obtained by solving the following differential equation,

∂ G C ˜ ( t 1 , T ) ∂ t 1 = 0 and ∂ G C ˜ ( t 1 , T ) ∂ T = 0 ,

And it should satisfy the condition

( ∂ 2 G C ˜ ( t 1 , T ) ∂ t 1 2 ) ( ∂ 2 G C ˜ ( t 1 , T ) ∂ T 2 ) − ( ∂ 2 G C ˜ ( t 1 , T ) ∂ t 1 ⋅ ∂ T ) > 0 .

Considering the above example for sensitivity analysis to study the effect of change in different parameters involved in the model. (Table 2)

1) As the value of η increases, Figure 2 and Figure 3 indicates that, value of t₁ & T decreases significantly but fuzzy total cost G C ˜ ( t 1 , T ) and Q increases.

2) As the value of β increases, Figure 4 and Figure 5 indicates that, value of t₁ & T increases and fuzzy total cost G C ˜ ( t 1 , T ) and Q decreases drastically.

3) As the value of γ increases, Figure 6 and Figure 7 indicates that, value of t₁, T and Q decreases significantly and fuzzy total cost G C ˜ ( t 1 , T ) increases.

4) As the value of λ increases, Figure 8 and Figure 9 indicates that, value of t₁, T and Q increases and fuzzy total cost G C ˜ ( t 1 , T ) decreases insignificantly.

5) As the value of t_m increases, Figure 10 and Figure 11 indicates that, value of t₁, T and Q increases insignificantly and fuzzy total cost G C ˜ ( t 1 , T ) decreases insignificantly.

It can be observed that, economic order quantity and fuzzy total cost is more sensitive towards demand coefficient and demand constant.

In this paper, the inventory model for deteriorating items deteriorates after a certain period of time and not instantaneously, where demand is dependent on selling price and shortages are allowed and fully backlogged. The total average cost for both crisp and fuzzy models is calculated. For the fuzzy inventory model, hexagonal fuzzy numbers are used and for defuzzification, the Graded mean integration representation method is used. By comparing the results of both models, the crisp and fuzzy model, it can be seen that a fuzzy model provides the optimum value of the total average cost.

In the modern industrialized era, the study emphasizes how important it is to adopt optimal inventory management procedures. Fuzzy inventory systems have demonstrated encouraging outcomes in terms of order quantity optimization, inventory cost reduction, and customer satisfaction. By taking into account different changes in demand patterns, this research adds a new dimension to the current understanding of inventory systems. It is anticipated that the developed fuzzy inventory systems would find practical usage in applications for businesses looking to maximize revenues while operating in uncertain environments.

In the future aspect, one can extend this paper by taking shortages 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. (2024) Fuzzy Inventory Model under Selling Price Dependent Demand and Variable Deterioration with Fully Backlogged Shortages. American Journal of Operations Research, 14, 87-103. https://doi.org/10.4236/ajor.2024.142005