An Optimum Inventory Policy for Exponentially Deteriorating Items , Considering Multi Variate Consumption Rate with Partial Backlogging

Abstract Customer purchasing deeds may be affected by factors such as selling price and inventory level instead of demand which is considered either constant or function of single variable which is not feasible. Consequently in the present study we have considered demand rate as a function of stock-level and selling price both. In present study, in order to develop this model it has been assumed that items are exponentially decaying and shortages are partially backlogged and most realistic backlogging rate is considered. In this research, we proposed a partial backlogging inventory model for exponentially decaying items considering stock and selling price dependent demand rate in fuzzy environment. In developing the model demand rate, ordering cost, purchasing cost, holding cost, backordering cost and opportunity cost are considered as triangular fuzzy numbers. Graded mean integration representation method is used for defuzzification. A numerical example is provided to illustrate the problem. Sensitivity analysis of the optimal solution with respect to the changes in the value of system parameters is also discussed.


INTRODUCTION
S everal items such as food items, vegetables and pharmaceuticals decay with time when kept in inventory.Therefore it is important to study the inventory of the items which deteriorate with time.Several authors have presented the inventory models of decaying items.Ghare & Schrader (1963) initially considered the impact of deterioration for a constant demand.Later on Shah & Jaiswal (1977), Aggarwal (1978), Covert & Philips (1973) and Goyal & Giri (2001) developed the inventory models of deteriorating item.Thus it is necessary to consider backlogging rate.Researchers, such as Park (1982), Hollier and Mak (1983) and Wee (1995) developed inventory models with partial backorders.Goyal and Giri (2003) developed production inventory model with shortages partially backlogged.Wu et al. (2006) developed a replenishment policy for deteriorating items with stock-dependent demand and partial backlogging.Singh (2008) presented a perishable inventory model with quadratic demand and partial backlogging.Skouri et al. (2009) presented an inventory model with ramp type demand rate, partial backlogging and Weibull deterioration rate.
The first model for decaying items was presented by Ghare and Schrader (1963).It was extended by Covert and Philip (1973) considering Weibull distribution deterioration.Goswami et al. (1991) developed an inventory model for deteriorating items with shortages and a linear trend in demand also a complete survey for deteriorating inventory models was presented by Raafat (1991).Some other papers relevant to this topic are Teng, et al., (2002), Chang et al.(2009) and Abdul and Murata, (2011).
Many classical research articles assumed that the demand is constant during the sales period.In real life, the demand may be inspired if there is a large pile of goods displayed on shelf.Levin et al. (1972) termed that the more goods displayed on shelf, the more customer's demand will be generated.Gupta and Vrat (1986) were the first to build up models for stock-dependent consumption rate.Baker and Urban (1988) established an EOQ model for a power-form inventory-level-dependent demand pattern.Mandal and Phaujdar (1989) then developed a production inventory model for deteriorating items with uniform rate of production and linearly stock-dependent demand.Other papers related to this research area are Giri and Chaudhuri (1998), Chung (2003), Chang (2004), Alfares (2007), Goyal and Chang (2009) and Chang et al. (2010).
In real life situations, due to impreciseness of parameters in inventory, it is important to consider them as fuzzy numbers.The concept of fuzzy set theory first introduced by Zadeh L. (1965), after that Park (1987) extended the classical EOQ model by introducing the fuzziness of ordering cost and holding cost.A fuzzy model for inventory with backorder, where the backorder quantity was fuzzified as the triangular fuzzy number was presented by Chang et al. (1998).Recently a supply chain inventory model under fuzzy demand was established by Ruoning Xu, Xiaoyan Zhai (2010).
Above cited papers reveals that many research articles are developed in which demand is considered as the function of stock level or selling price, shortages are allowed and partially backlogged, but there is no such research paper which is partially backlogged assuming demand rate as the function of selling price with exponentially decaying and inventory level in fuzzy environment.In lots of business practices it is observed that several parameters in inventory system are imprecise.Therefore, it is necessary to consider them as fuzzy numbers while developing the inventory model.In the present study we have developed a partial backlogging inventory model for exponential deteriorating items considering stock and price sensitive demand rate in fuzzy surroundings.A numerical example to prove that the optimal solution exists and is unique is provided and the sensitivity analysis with respect to system parameters is discussed.The concavity is also shown through the figure.

II. ASSUMPTIONS AND NOTATIONS
The basic assumptions of the model are as follows: (1) The demand rate is a function of stock and selling price considered as f (t) = (a+bQ(t) − p) where a > 0, 0 < b < 1, a > b and p is selling price.(2) Holding cost h(t) per item per time-unit is time dependent and is assumed to be h(t) = h +δt where δ > 0, h > 0.
(3) Shortages are allowed and partially backlogged and rate is assumed to be 1/ (1+ηt) which is a decreasing function of time.(4) The deterioration rate is time dependent.
(5) T is the length of the cycle.(6) Replenishment is instantaneous and lead time is zero.(7) The order quantity in one cycle is Q. (8) The selling price per unit item is p. (9) A is the cost of placing an order.(10) c 1 is the purchasing cost per unit per unit.(11) c 2 the backorder cost per unit per unit time.(12) c 3 the opportunity cost (i.e., goodwill cost) per unit.(13) P(T,t 1 ,p) the total profit per unit time.( 14) The deterioration of units follows the exponential distribution (say) Where F(t) is distribution function of exponential distribution.(15) During time t1, inventory is depleted due to deterioration and demand of the item.At time t1 the inventory becomes zero and shortages start going on.

MATHEMATICAL FORMULATION AND SOLUTION
Let Q (t) be the inventory level at time t (0 ≤ t ≤ T).During the time interval [0, t 1 ] inventory level decreases due to the combined effect of demand and deterioration both and at t 1 inventory level depletes up to zero.The differential equation to describe immediate state over [0, t 1 ] is given by Again, during time interval [t 1 ,T] shortages stars occurring and at T there are maximum shortages, due to partial backordering some sales are lost.The differential equation to describe instant state over [t 1 ,T] is given by With condition Solving equation (1) and equation ( 2) and neglecting higher powers of α, we get At time 0 inventory level is Q(0) and is given by At time T maximum shortages (Q1) occurs and is given by The order quantity is Q and is given by The purchasing cost is Ordering cost is Shortage cost due to backordered is Lost sales cost due to lost sales is Sales revenue is given by

SR p a bQ t p dt p a p dt p a p T b a p p t bt
From ( 6), ( 7), ( 8), ( 9) and (10) total profit per unit time is given by Let t 1 = γT, 0 < γ < 1 Hence we get the profit function software Mathematica-8.1, from equation ( 13) and ( 14) we can determine the optimum values of T * and p * simultaneously and the optimal value P*(T, P) of the average net profit is determined by (12) provided they satisfy the sufficiency conditions for maximizing P * (T, p) are

IV. OBSERVATIONS
It is observe from the table that optimal replenishment quantity and total profit increases as the parameters a, b increases.As the parameter c 1 increases the order quantity increases but the total profit slightly decreases.The optimal replenishment quantity and total profit decreases as the parameters c 2 , λ, η and h increases.The optimal order quantity and total profit very slightly decreases as the parameter δ increases.

V. FUZZY MATHEMATICAL MODEL
In this study we consider a, A, c 1 , c 2 , c 3 , h and δ as fuzzy numbers i.e, as Then ˆ, ˆ, ˆ, ˆ, ˆ, ˆâ A c c c h 1 2 3 and δ P * (T,P) is regarded as the estimate of total profit per unit time in the fuzzy sense.
Where 0 < a and Wh here 0 < A and Where 0 < Where 0 < c and Where 0 < c and And the signed distance of â to 0 is given by the relation Similarly, the signed distance of other parameters to 0 is given by the relations Where d ( ) > 0 and d( Where d ( Where d ( ) > 0 and d( Now, by the fuzzy triangular rule fuzzy total profit per unit is Now, defuzzified average profit is given by Pandey, H. Pandey, Also,the defuzzified order quantity is Q and is given by The necessary conditions for maximizing the average profit are

VIIII. CONCLUSION
In the current study an inventory model is presented in which demand rate is considered as a function of price and stock both and exponentially decaying.In the development of model it is assumed that shortages are allowed and partially backlogged.The model is proposed in the following two senses: (1) crisp sense and (2) fuzzy sense.A numerical example to illustrate the problem in both the environments is provided and sensitivity analysis with respect to system parameters is also carried out.

Figure 1 :
Figure 1: Concavity of the profit function.

Figure 2 :
Figure 2: Net Profit v/s change in parameter a.

Figure 3 :
Figure 3: Net Profit v/s change in parameter b.

Figure 4 :
Figure 4: Net Profit v/s change in parameter c 1.

Figure 5 :
Figure 5: Net Profit v/s change in parameter c 2 .

Figure 6 :
Figure 6: Net Profit v/s change in parameter λ.

Figure 7 :Figure 8 :
Figure 7: Net Profit v/s change in parameter h

Figure 9 :
Figure 9: Net Profit v/s change in parameter δ.