Fuzzy Inventory Model for Deteriorating Items with Time Dependent Demand and Partial Backlogging

In this paper we developed a fuzzy inventory model for deteriorating items with time dependent demand rate. Shortages are allowed and completely backlogged. The backlogging rate of unsatisfied demand is assumed to be a decreasing exponential function of waiting time. The demand rate, deterioration rate and backlogging rate are assumed as a triangular fuzzy numbers. The purpose of our study is to defuzzify the total profit function by signed distance method and centroid method. Further a numerical example is also given to demonstrate the developed crisp and fuzzy models. A sensitivity analysis is also given to show the effect of change of the parameters.


Introduction
In many inventory models uncertainty is due to fuzziness and fuzziness is the closed possible approach to reality.In recent years some researchers gave their attention towards a time dependent rate because the demand of newly launched products such as fashionable garments, electronic items, mobiles etc. increases with time and later it becomes constant.Deterioration is defined as damage, decay or spoilage of the items that are stored for future use always loose part of their value with passage of time, so deterioration cannot be avoided in any business scenarios.F. Harris (1915) [1] developed first inventory model.Lotfi A. Zadeh (1965) [2] introduced the concept of fuzzy set theory in inventory modeling.L. A. Zadeh [3] and R. E. Bellman (1970) considered an inven-tory model on decision making in fuzzy environment.R. Jain (1976) [4] developed a fuzzy inventory model on decision making in the presence of fuzzy variables.D. Dubois and H. Prade (1978) [5] defined some operations on fuzzy numbers.J. Kacpryzk and P. Staniewski (1982) [6] developed an inventory model for long term inventory policy making through fuzzy decisions.H. J. Zimmerman (1983) [7] tried to use fuzzy sets in operational research.G. Urgeletti Tinarelli (1983) [8] considered the inventory control models and problems.K. S. Park (1987) [9] define the fuzzy set theoretical interpretation of an EOQ problem.M. Vujosevic, D. Petrovic and R. Petrovic (1996) [10] developed an EOQ formula by assuming inventory cost as a fuzzy number.J. S. Yao and H. M. Lee (1999) [11] developed a fuzzy inventory model by considering backorder as a trapezoidal fuzzy number.C. K. Kao and W. K. Hsu (2002) [12] developed a single period inventory model with fuzzy demand.C. H. Hsieh (2002) [13] developed an inventory model and give an approach of optimization of fuzzy production.J. S. Yao and J. Chiang (2003) [14] developed an inventory model without backorders and defuzzified the fuzzy holding cost by signed distance and centroid methods.Sujit D. Kumar, P. K. Kund and A. Goswami (2003) [15] developed an economic production quantity model with fuzzy demand and deterioration rate.J. K. Syed and L. A. Aziz (2007) [16] consider the signed distance method for a fuzzy inventory model without shortages.P. K. De and A. Rawat (2011) [17] developed a fuzzy inventory model without shortages by using triangular fuzzy number.C. K. Jaggi, S. Pareek, A. Sharma and Nidhi (2012) [18] developed a fuzzy inventory model for deteriorating items with time varying demand and shortages.
Sumana saha and Tripti Chakrabarty (2012) [19] developed a fuzzy EOQ model with time varying demand and shortages.D. Dutta and Pawan Kumar (2012) [20] considered a fuzzy inventory model without shortages using a trapezoidal fuzzy number.D. Dutta and Pawan Kumar (2013) [21] [22] considered an optimal replenishment policy for an inventory model without shortages by assuming fuzziness in demand, holding cost and ordering cost.Dipak Kumar Jana, Barun Das and Tapan Kumar Roy (2013) [23] give a fuzzy generic algorithm approach for an inventory model for deteriorating items with backorders under fuzzy inflation and discounting over random planning horizon.
In this paper we consider an inventory model for deteriorating items with time dependent demand rate and partial backlogging.Shortages are allowed and completely backlogged for the next replenishment cycle.The demand rate, deterioration rate and backlogging rate are assumed as triangular fuzzy numbers.The purpose of our study is to defuzzify the total profit function by signed distance method and centroid method and comparing the results of these two methods with the crisp model.Figure 1 shows the developed model and Figure 2 and Figure 3 show the graphs of total profit function with respect to deterioration and backlogging rates.

Definitions and Preliminaries
When we are considering the fuzzy inventory model then the following definitions are needed.
(1) A fuzzy set A  on the given universal set X is denoted and defined by ( )   And, ( ) (2) A fuzzy number A  is a fuzzy set on the real line R , if its membership function A λ  has the following properties is a triangular fuzzy number then the signed distance of A  is defined as is a triangular fuzzy number then the centroid method on A  is defined as

Assumptions and Notations
We consider the following assumptions and notations.
The demand rate is ( ) where α is a positive constant, for a increasing demand 1 β > , and for a decreasing demand 1 β < . 1. θ is the deterioration parameter. 2. σ is the backlogging parameter. 3. A is the ordering cost per order.4. c h is the holding cost per unit per unit time. 5. c d is the deterioration cost per unit per unit time.6. c s is the shortages cost per unit per unit time.7. c p is the purchase cost per unit.8. p s is the selling price per unit, where p C s p  .9. pC o is the opportunity cost per unit due to lost sales. 10.T is the length of order cycle. 11. θ  is the fuzzy deterioration parameter. 12. σ is the fuzzy backlogging parameter.
13. α  is the fuzzy demand parameter.
14.  ( ) 1 , TP T T is the total fuzzy profit per unit time.
15. 1 T is the time at which shortage starts.16.

( )
1 , TP T T is the total profit per unit time.

Mathematical Formulation
Suppose an inventory system consists Q units of the product in the beginning of each cycle.Due to demand and deterioration the inventory level decreases in [ ] 1 0,T and becomes zero at T T is the shortages interval.During the shortages interval the unsatisfied demand is backlogged at a rate of e t σ − , where t is the waiting time.
The instantaneous inventory level at any time t in [ ] 1 0,T are governed by the following differential equa- tions with boundary condition ( ) with boundary condition ( ) The solution of Equation ( 1) is The solution of Equation ( 2) is ( ) ( ) ( ) using ( ) The ordering cost per cycle is The holding cost per cycle is The deterioration cost per cycle is The shortage cost per cycle is The purchase cost per cycle in [ ] The purchase cost per cycle in [ ] .
Due to lost sales the opportunity cost per cycle in [ ] ( ) ( ) . 1 The sales revenue cost per cycle in [ ] 0,T is ( ) . 1 1 Therefore the total profit per unit time is For a Ist order approximation of e t θ − ( ) ( ) ) The necessary condition for ( ) 1 , TP T T to be maximum is that ( ) ( ) , and solving these equations we find the optimum values of 1 T and T say 1 T * and T * for which profit is maxi- mum and the sufficient condition is ( ) ( ) 1 , ,

Fuzzy Model
Let us consider the inventory model in fuzzy environment due to uncertainty in parameters let us assume that the parameters θ , α and σ may change within some limits.Let ( ) , , , , ( ) , , σ σ σ σ = are triangular fuzzy numbers then the total profit per unit time in fuzzy sense is Now we defuzzify the total profit  ( ) 1 , TP T T in two cases.

Signed Distance Method
By signed distance method the total profit per unit time is  ( )  ( )  ( )  ( )

TP T T TP T T TP T T TP T T T
where, ) ) The necessary condition for  ( ) 1 , TP T T to be maximum is that , and solving these equations we find the optimum values of 1 T and T say 1 T * and T * for which profit is maxi- mum and the sufficient condition is ( )

Centroid Method
By Centroid method the total profit per unit time is

T T TP T T TP T T TP T T T
) The necessary condition for  ( ) 1 , TP T T to be maximum is that , and solving these equations we find the optimum values of 1 T and T say 1 T * and T * for which profit is maxi- mum and the sufficient condition is  ( ) ( ) ( )

Numerical Example
Let us consider an inventory system with the following parameters in appropriate units as Rs200/order A = Rs20/unit/year .
Table 1 shows that as we increase deterioration parameter θ then the total profit increases.
Table 2 shows that as we increase backlogging parameter σ then the total profit increases.Table 3 shows that as we increase demand parameter α then the total profit increases.The solution of the fuzzy inventory model can be determined by the following two methods.

Signed Distance Method
When α , σ and θ are triangular fuzzy numbers, then Table 4 shows the value of total profit.When α and σ are triangular fuzzy numbers, then Table 5 shows the value of total profit.When α and θ are triangular fuzzy numbers, then Table 6 shows the value of total profit.

Centroid Method
When α , σ and θ are triangular fuzzy numbers, then Table 7 shows the value of total profit.When α and σ are triangular fuzzy numbers, then Table 8 shows the value of total profit.When α and θ are triangular fuzzy numbers, then Table 9 shows the value of total profit.

Sensitivity Analysis
From Table 1, we see that as we increase the deterioration parameter θ then the optimal time period 1 T , the optimal cycle time T and total profit increases.
From Table 2, we see that as we increase the backlogging parameter σ then the optimal time period 1 T , the optimal cycle time T decreases and total profit increases.
From Table 3, we see that as we increase the demand rate parameter α then the optimal time period 1 T decreases and the optimal cycle time T and total profit increases.
In the case of crisp model we see that the backlogging parameter σ is more sensitive than the deterioration parameter θ and the demand rate parameter α .
From the tables for signed distance method and centroid method we see that the fuzzy variables θ  and α  are more sensitive than the fuzzy variable σ .As we increase the fuzzy variables θ  and α  in the signed distance method and centroid method than the total profit increases rapidly in centroid method.Therefore in the sense of fuzziness the centroid method is better one than the signed distance method.

Conclusion
In this paper we studied a fuzzy inventory model for deteriorating items with time dependent demand rate and partial backlogging.Shortages are allowed and completely backlogged.As we increase the parameters α , σ and θ in the crisp model then the total profit increases and due to the uncertainties in the demand rate, deteri- oration rate and backlogging rate the parameters α , σ and θ are consider as triangular fuzzy numbers.For defuzzification by signed distance method and centroid method it has been observed that the total profit decreases as the optimal cycle time decreases and the profit given by the signed distance method is minimum as compared to the centroid method.Further this model can be generalized by considering time dependent deterioration rate, holding cost, shortage cost and so many types.

Figure 1 .
Figure 1.With respect to described model.

Table 1 .
Variation in total profit with respect to θ .

Table 2 .
Variation in total profit with respect to σ .

Table 3 .
Variation in total profit with respect to α .

Table 5 .
Variation in total profit with fuzzy numbers, α and σ .

Table 6 .
Variation in total profit with fuzzy numbers, α and θ .

Table 8 .
Variation in total profit with fuzzy numbers, α and σ .

Table 9 .
Variation in total profit with fuzzy numbers, α and θ .