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In this paper, an attempt has been made to develop a simple leakage inventory model without shortages with instantaneous or finite production rate under fuzzy environment. In the present day scenario, it is difficult to decide the exact annual demand rate and hence a major difficulty is faced by a decision maker to forecast the demand. Also, in any inventory system, goods in stock may subject to deterioration or leakage. Deterioration of goods refers to decrease in quality whereas quantity remains more or less the same. On the other hand, leakage refers to loss in quantity whereas the quality remains unchanged for a certain period of time. Leakages in the inventory system may be considered to be very small, not detectable by the management immediately and hence it is difficult to decide the exact leakage rate. The objective of this paper is to consider these variable parameters and determine the optimal economic order quantity (EOQ) to maximize the annual total profit. So, fuzzy inventory models have been proposed considering fuzzy leakage rate and fuzzy annual demand rate to estimate the total profit per unit time. Signed distance method is used for defuzzification. Numerical examples are provided to support the results of the proposed models.

The present study pertains to a mathematical model that can be implemented for the successful running of a factory of production facility where there may be loss of resources/raw materials due to leakage is the storage facility involved. Understanding such associated problem will definitely help in the optimization of operations thus maximizing profitability of the entire establishment.

Taleizadeh [1] considered an EOQ model with partial backordering and advance payments for an evaporating item. So, considering leakage as analogous to deterioration, Tomba and Geeta [2] developed a leakage inventory model having no shortages with uniform demand rate and instantaneous production rate.

In developing the models, there are many assumptions and parameters which are considered to be fixed and have exact values. But in reality, these assumptions seem to be unrealistic since they are generally vague and imprecise, sometimes even it is impossible to determine the exact value. In this context, fuzzy inventory models are discussed as an extension of the traditional or crisp inventory models. Yao and Lee [3] solved the inventory model with shortages by fuzzifying the order quantity using extension principle. Lee and Yao [4] also discussed production inventory problems by defuzzifying demand quantity and production quantity considering triangular fuzzy numbers. Chang [5] developed a fuzzy production inventory for fuzzy product quantity using a triangular fuzzy number. Yao and Su [6] studied the fuzzy inventory model with backorder for fuzzy total demand based on interval-valued fuzzy set. Chang [7] discussed the EOQ model with imperfect quality items by applying the set theory considering fuzzy defective rate and fuzzy annual demand. Mahata [8] applied fuzzy set theory in an EOQ model for items with imperfect quality and shortage backordering. Jaggi et al. [9] studied a fuzzy inventory model for deteriorating items with initial inspection and allowable shortage under the condition of permissible delay in payments. Patro et al. [10], in their paper, discussed an EOQ model for fuzzy defective rate with allowable proportionate discount. Chen and Ouyang [11], in their paper extended an ordering policy for deteriorating items with allowable shortage and permissible delay in payment to fuzzy model by fuzzifying the carrying cost, interest paid rate and interest earned rate simultaneously based on the interval-valued fuzzy numbers and triangular fuzzy numbers. Chang et al. [12] discussed a fuzzy mixed inventory model involving variable lead-time with backorder and lost sales using probabilistic fuzzy set and triangular fuzzy number. They used two methods of defuzzification. In this paper, we investigate the leakage inventory model based on the Tomba and Geeta  model (TG-model) [2] incorporated with a fuzzy leakage model [13] by considering demand rate and leakage rate as triangular fuzzy numbers and the model is used to find the optimal order quantity and the minimum total cost per unit time in fuzzy environment.

1. Theoretical basis of the model :

Tomba and Geeta [2] developed a deterministic leakage inventory model without shortages having uniform demand rate with instantaneous production. They assumed the lead time of an order quantity q to be zero in developing their model. According to their model, total holding cost =  htq/2, where h and q denote the holding cost per unit quantity per unit time t and order quantity per run respectively.

Fig. 1. Instantaneous production leakage inventory without shortage.

2. Definitions and Preliminaries

To extend the TG- model to fuzzy environment, the following definitions and preliminaries are taken into account:

2.1    Fuzzy point

2.2    α-level fuzzy point

2.3 Triangular fuzzy number

2.4 α-level fuzzy interval

2.5 α-cut of a fuzzy set

2.6  Decomposition principle

2.7          Signed distance (as in Yao and Wu [13])

3. Fuzzy leakage inventory model

4. Numerical example and sensitivity analysis

To illustrate the applicability of the model, the example from Tomba and Geeta [2] is considered.

The crisp inventory system has the data: demand rate ϕ=600 units per unit per year, holding cost h = Rs10, setup cost s = Rs100 per order, leakage rate ψ = 10 units per unit per year. Then, the optimal order quantity q* = 108 and the minimum cost per year C_min= Rs1104.

In our fuzzy model, instead of taking the demand rate and leakage rate as fixed quantity, we assumed them to be imprecise and hence represented by triangular fuzzy numbers  = (600−Δ₁, 600, 600+Δ₂) and  = (10−Δ₃, 10, 10 + Δ₄) respectively in accordance to our fuzzy model. The optimal order lot size q* and minimum total cost Z* for various sets of Δ₁, Δ₂, Δ₃, Δ₄ are summarized in Table 1. Relative variation between fuzzy case and crisp case for the optimal lot size q* and minimum total cost Z* is also calculated as and respectively, where q*and Z* denote the optimal order quantity and minimum total cost in crisp sense respectively.

Table 1. Optimal Solution for the proposed model with fuzzy demand rate and fuzzy leakage rate.

Δ1	Δ2	Δ3	Δ4	d(ϕ, 0)		q*	Z*	Relq	RelZ
100	100	1	1	600	0.00162	109.456	1099.76	1.3478	-0.6943
			2	600	-0.00095	109.597	1094.92	1.4783	-0.8221
		4	3	600	-0.01199	110.207	1088.86	2.0436	-1.3715
			4	600	-0.01440	110.342	1087.53	2.1686	-1.4921
		7	5	600	-0.02498	110.939	1081.67	2.7215	-2.0223
			6	600	-0.02725	111.068	1080.42	2.8410	-2.1362
150	200	1	1	612.5	0.00395	110.462	1108.98	2.2792	0.4514
			2	612.5	0.00230	110.553	1108.07	2.3636	0.3686
		4	3	612.5	-0.00385	110.893	1104.67	2.6790	0.0603
			4	612.5	-0.00544	110.982	1103.78	2.7613	-0.0199
		7	5	612.5	-0.01142	111.317	1100.46	3.0717	-0.3209
			6	612.5	-0.01296	111.404	1099.60	3.1520	-0.3986
200	300	1	1	625	0.00496	111.527	1120.80	3.2659	1.5221
			2	625	0.00378	111.592	1120.15	3.3263	1.4627
		4	3	625	-0.00052	111.832	1117.75	3.5484	1.2451
			4	625	-0.00166	111.896	1117.11	3.6077	1.1871
		7	5	625	-0.00587	112.133	1114.75	3.8268	0.9736
			6	625	-0.00698	112.196	1114.12	3.8850	0.9170

From the above Table 1, we observed that

1. for fixed Δ₁ and  Δ₂, the demand rate is constant and with the increase in variation of Δ₃ and Δ₄,the  optimal lot size q* increases and minimum total cost Z* decreases. Hence, Rel_q  increases and Rel_Z decreases.

2. for fixed Δ₃ and Δ₄, both the optimal lot size q* and minimum total cost Z* increases and hence both Rel_q and Rel_Z increases.

From the discussed example and the sensitivity analysis of the results from Table 1, we cannot exactly ascertain the better results of the model in the two different environments. But still, the one in fuzzy sense will be more reliable as it incorporates real-life situations and conditions with wide range of variability.

In this paper, we have discussed a leakage inventory model in fuzzy environment and compared it with its traditional model in crisp sense by considering different data. The main parameters of the model, namely the demand rate and the leakage rate are assumed to be triangular fuzzy numbers. The optimum results are obtained by using signed distance method of defuzzification. It is observed that for different sets of fuzzy demand rate and fuzzy leakage rate, the optimum ordering quantity and minimum total cost are almost more or less with that obtained in crisp environment. Also, uncertainties that are prevalent in real inventory problems are highlighted in this fuzzy model and from the sensitivity analysis by taking different values of Δ1, Δ2, Δ3 and Δ4 the variations or the effect of uncertainties the optimum ordering quantity and minimum total cost are analyzed.

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