A Warehouse Imperfect Fuzzified Production Model with Shortages under Inflationary Conditions

We develop a two-warehouse production model with imperfect items. Production rate is taken as the linear combination of on-hand inventory and demand, while demand rate is taken as function of time. Most of the researchers consider that the production rate is independent from the demand rate. In this paper we assume production rate as being dependent on the demand rate, and this assumption is more realistic. Shortages are allowed and partially backlogged with time-dependent backlogging rate. Due to different preservation facilities we consider that the deterioration rate is time dependent in own warehouse OW and Weibull distribution deterioration in rented warehouse RW . Holding cost in RW is greater than in OW. We developed a fuzzy model with fuzzifying all the costs of the model as triangular fuzzy numbers. The present model is developed in both crisp and fuzzy senses. Finally, numerical example is shown, and sensitivity is also illustrated.


Introduction
One of the weaknesses of some production-inventory models is the unrealistic assumption that all items produced are of good quality. But production of defective units is a natural phenomenon in a production process. Defective items should be treated as a result of imperfect quality production. The effect of an imperfect process on production run time and EPQ was initially studied by Rosenblatt and Lee 1 . In their study, the elapsed time until the process shift was assumed to be exponentially distributed. The optimal production run was found to be shorter than that of classical EPQ model. In recent years, numerous research efforts have been undertaken to extend the work of Rosenblatt and Lee 1 . Kim and Hong 2 extended the work of Rosenblatt and Lee 1 by assuming that elapsed time 2 Advances in Decision Sciences until the process shift was arbitrarily distributed. However, neither of their models took into consideration of any allowable shortages. Chung and Hou 3 , however, have generalized the work of Kim and Hong 2 by assuming that shortages were allowed. Nevertheless, the common assumption of all the above-mentioned models was that there were a fixed percentage of defective items produced during the out-of-control period. C. Singh and S. R. Singh 4 developed an imperfect production process with exponential demand rate, Weibull deterioration under inflation.
In some cases, uncertainties can be defined as fuzziness or vagueness, which are characterized by fuzzy numbers of the fuzzy set theory. Zadeh 5 introduced fuzzy set theory to deal with quality-related problems with imprecise demand. Bellman and Zadeh 6 distinguished the difference between randomness and fuzziness by showing that the former deals with uncertainty regarding membership or nonmembership of an element in a set, while the later is concerned with the degree of uncertainty by which an element belongs to a set. Gen et al. 7 expressed the input data by fuzzy numbers, where they used interval mean value concept to solve an inventory problem. Yao and Chiang 8 considered an inventory model with total demand and storing cost as triangular fuzzy numbers. They performed the defuzzification by centroid and signed distance methods. Mondal and Maiti 9 applied genetic algorithms GAs to solve a multi-item fuzzy EOQ model. M. K. Maiti and M. Maiti 10 dealt with a fuzzy inventory model with two warehouses under possibility constraints. Mahapatra and Maiti 11 formulated a multi-item, multiobjective inventory model for deteriorating items with stock-and time-dependent demand rate over a finite time horizon in fuzzy stochastic environment. Halim et al. 12 developed a fuzzy inventory model for perishable items with stochastic demand, partial backlogging, and fuzzy deterioration rate. The model is further extended to consider fuzzy partial backlogging factor. Gani and Maheswari 13 discussed the retailer's ordering policy under two levels of delay payments considering the demand and the selling price as triangular fuzzy numbers. They used graded mean integration representation method for defuzzification. Lee and Yao 14 developed an economic production quantity EPQ model in which the demand and the production quantity are assumed to be fuzzy. Halim et al. 15 addressed the lot sizing problem in an unreliable production system with stochastic machine breakdown and fuzzy repair time. They defuzzified the cost-per-unit time using the signed distance method. Production of defective items in any manufacturing industry is a natural phenomenon. The number of defectives may have a change from one lot to another that cannot be assessed by a crisp value. Chen and Chang 16 developed a fuzzy economic production quantity EPQ model with defective productions that cannot be repaired. In this model, they considered a fuzzy opportunity cost and trapezoidal fuzzy costs under crisp production quantity or fuzzy production.
Since most decision makers think that the inflation does not have significant influence on the inventory policies, the effects of inflation are notconsidered in some inventory models. However, from a financial point of view, an inventory represents a capital investment and must compete with other assets for a firm's limited capital funds. Thus, it is necessary to consider the effects of inflation on the inventory system. Buzacott 23,24 are amongst the few who studied the concept of inflation with regard to inventory. Singh and Jain 25 explored a deterministic inventory model for a deteriorating item in an inflation-induced environment.

3
In this paper, we assumed the following assumptions.
i Production rate is greater than the demand rate.
ii Production rate is the linear combination of on-hand inventory and demand rate.
iii Demand rate is exponentially an increasing function of time.
iv Model is considered for imperfect items.
v Deterioration is taken as time dependent for OW, while, Weibull distribution for RW.
vi Inflation is also taken in this model.
vii Costs are considered as a triangular fuzzy numbers.
viii Model is presented in both fuzzy and crisp senses.

Assumptions
The following assumptions have been used throughout the study.
i The demand rate is exponentially increasing and is represented by λ t λ 0 e δt , where 0 ≤ δ ≤ 1 is a constant inflation rate, and λ 0 is the initial demand rate.
ii Lead time is zero, and no replenishment or repair of deteriorated items is made during a given cycle.
iii Production rate is the linear combination of on-hand inventory and demand rate at any time subject to P t I t ηλ t 1 − e −Δt , 0 ≤ Δ ≤ 1, η > 1, P t > λ t .

2.1
iv Shortages are allowed and the backlogging rate is exponentially, a decreasing function of time when inventory is in shortages, the backlogging parameter s is a positive constant s.t. 0 < s < 1.
v A single item is considered over the prescribed period T units of time, which is subject to variable deterioration rate.
vi The owned warehouse OW has a fixed capacity of W units, and the rented warehouse RW has unlimited capacity.
vii The goods of OW are consumed only after consuming the goods kept in RW.
viii The unit inventory costs including holding cost per unit time in RW are higher than those in OW.
ix Deterioration-rate of the items is considered to be different in different warehouses.
x The model is developed in both fuzzy and crisp environment.
xi Finite planning horizon is considered.

Notations
The following notations have been used throughout the study.
λ: Demand rate, λ t λ 0 e δt , increases with time, where λ 0 is the initial demand rate and 0 ≤ δ ≤ 1. P t : Production rate P t I t ηλ t 1 − e −Δt , which is the linear combination of on-hand inventory and demand rate at any time and P t > λ t , where 0 ≤ Δ ≤ 1, η > 1.       I i2 : Inventory level in RW at time t with t ∈ t 1 , t 2 .
I i3 : Inventory level in RW at time t with t ∈ 0, t 3 .
I i4 : Inventory level in OW at time t with t ∈ 0, t 4 .
I i5 : Inventory level in OW at time t with t ∈ 0, t 2 t 3 .
I i6 : Inventory level in OW at time t with t ∈ 0, t 5 .
I i7 : Inventory level in OW at time t with t ∈ 0, T .

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Inventory level Figure 1: Inventory level in production system for deteriorating items with inflation-induced demand. Figure 1 shows the inventory level during a production cycle in which both OW and RW are used. Initially, the inventory level is zero. The production starts at time t 0, and items accumulate from 0 up to W units in OW and in t 1 units of time. After time t 1 any production quantity exceeding W will be stored in RW. After this production stopped and the inventory level in RW begins to decrease at t 2 and will reach 0 units at t 3 because of demand and deterioration. The inventory level in OW comes to decrease at t 1 and then falls below W at t 2 t 3 due to deterioration. The remaining stocks in OW will be fully exhausted at t 4 owing to demand and deterioration: the inventory becomes zero. At this time shortage starts developing, and at time t 5 it reaches to maximum shortage level : at this time fresh production starts to clear the backlog by the time T . The differential equations stating the inventory levels within the cycle are given as follows:

Formulation and Solution of the Model
With the boundary conditions I i1 0 0, I i2 t 1 0, I i3 t 3 0, I i4 t 4 0, I i5 t 1 W, I i6 t 4 0, and I i7 T 0, respectively, the above equations can be solved successively as follows: 3.14 Therefore, we find out some costs such as the following.

Present Worth Ordering Cost
Consider the following: Ordering cost per cycle C 1 e −rt 5 . 3.15

Present Worth Inventory Level in RW
The total inventory level in RW can be derived as 2 .

3.16
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Present Worth Inventory Level in OW
The total inventory level in OW can be derived as 3.17

Present Worth Deteriorated Items
The total quantity of deteriorated items during the period 0, T is given by

Present Worth Shortage Cost
Total quantity of shortage units I S during the period 0, T is given by 3.19

Present Worth Lost Sales Quantity
The total amount of lost sales I L during the period 0, T can be obtained as

3.20
Number of defective items is 3.21 Expected number of defective items in a production cycle is f X is probability density function of X: Advances in Decision Sciences 9

Present Worth of Rework Cost
The total amount of rework cost I R during the period 0, T 2 can be obtained as 3.24

Present Worth of Total Cost
The total average cost TC of the inventory system is given by 3.25

4.1
By Centroid Method consider

4.2
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Numerical Example
To illustrate the theory of the model we consider the following data on the basis of the previous study.

Sensitivity Analysis
See Table 1.

Observations
1 As we increase the ordering cost, the optimum value of t * 1 , t * 3 , t * 4 , t * 5 decreases, and t * 2 remains same. By this effect, our total cost decreases.
2 As we increase the deterioration parameter for OW, the optimum value of t * 1 , t * 2 increases and that of t * 3 , t * 4 , t * 5 decreases, so our total cost decreases.
3 As we increase the deterioration parameters for RW, the optimum value of t * 1 , t * 4 , t * 5 , increases and that of t * 2 , t * 3 decreases then total cost slightly increase decreases. So the total cost is minimum when the deterioration rate of own warehouse is less than the deterioration rate of rented warehouse.

Conclusion
This paper presented a warehouse imperfect production-inventory models for deteriorating items having time varying demand patterns with Weibull distribution deterioration and partial backlogging under the effect of inflation and time value of money. Inflation plays a very significant role in the inventory models. Demand, production, defective item production, and so forth are natural in any manufacturing system. In this paper, we have developed two models for a manufacturing system. In the first model, all costs were assumed to be stochastic whereas in the second model, the costs were assumed to be fuzzy. The production rate was proportional to the demand rate. Well-known triangular membership function was used for all the fuzzy numbers. Numerical examples are solved by the software MATHEMATICA 8.0. Convexity of the total average cost is also illustrated in Figure 2. However, the advantage of the fuzzy approach is that it relaxes the assumptions such as defective rate, demand rate etc. Also, it eases the difficulties in searching for suitable probability distribution function to represent the random behavior of uncontrollable variables.