Inventory Replenishment Policy with Time and Reliability Varying Demand

Abstract: In real life situation, demand may be increase, decrease or constant. But, in our present study demand is assumed to be an increasing function of time which depends on reliability. Shortages are allowed and excess demand is backlogged. The economic production lot size and the reliability of the production process along with the production period are the decision variables and total cost per cycle is the objective function which is to be minimized. Further the parameters involved in the business may likely to be changed due to the fast growing marketing system. Therefore; it will be more realistic and market friendly to deal with a fuzzy model rather than a crisp model. Both crisp and fuzzy models have been proposed to determine the optimal solution. The demand, shortage cost, holding cost and deterioration rate and reliability are considered as pentagonal fuzzy numbers. Defuzzification of the total cost has been carried out by Graded Mean Representation Method and Signed Distance Method. Sensitivity analysis is also incorporated to investigate the effect of different system parameters in enhancing the cost.


I. INTRODUCTION
Inventory control is the supervision of supply, storage and accessibility of items in order to ensure an adequate supply without excessive oversupply. It can also be referred as the process of managing the timing and the quantities of goods to be ordered and stocked, so that demands can be met satisfactorily and economically. Success of inventory control depends on some important issues, i.e., uncertainty about the size of future demands, uncertainty of inventory cost, uncertainty of deterioration, reliability of the production process, etc. In the above cases, it has been assumed that the inventory parameters are crisp or precise or probabilistic but in reality they may deviate a little from their actual value without following any probability distribution. To deal with such type of uncertainty in inventory parameters, the notion of fuzziness has been initialized by Zimmermann [4]. This model has also been maneuvered by various researchers. Tripathy et al. established a fuzzy Economic Order Quantity model with reliability where the unit cost depends on demand [3]. Dutta and Kumar developed a fuzzy inventory model without shortage where holding cost and ordering cost were taken as trapezoidal fuzzy numbers [1]. Tripathy and Pattnaik focused on optimal disposal mechanism by considering the system cost as fuzzy under flexibility and reliability criteria [5]. Tripathy and Behera formulated a fuzzy inventory model for time deteriorating items using penalty cost under the condition of infinite production rate [6]. Dutta and Kumar explored an optimal ordering policy for an inventory model for deteriorating items without shortage where demand rate, ordering cost and holding cost were taken as fuzzy in nature [2]. Jaggi et al. introduced a fuzzy inventory model for deteriorating items with time varying demand and shortages [8]. Singh and Rathore studied an inventory model for deteriorating items with reliability consideration and trade credit [7].
is a pentagonal fuzzy number then the signed distance of C is defined as

Arithmetic Operations of PFN:
Formation of an arithmetic operation is crucial in the study of fuzzy numbers; the author tries to establish some basic arithmetic operations of PFN.

III. ASSUMPTIONS AND NOTATIONS
We adopt the following assumptions and notations for the models to be discussed.

Assumptions:
is assumed to be an increasing (iv) Shortages are allowed and fully backlogged.

Notations:
(i)   t R : Demand rate for any time per unit time. (ii) 0 A : Ordering cost per order.
(iv) T : Length of the cycle.
(v) 0 Q : Ordering quantity per unit time.
(viii) 0 C : Unit cost per unit time.
(ix) r : Reliability of the product. (xiii) 0 h : Fuzzy holding cost per unit time.
(xiv) 0 S : Fuzzy shortages cost per unit time.
(xv) 0 C : Fuzzy unit cost per unit time.
(xvi)   is the period of shortages, which are fully backlogged. Then the behavior of the inventory level is governed by the following differential equations.

Crisp Model:
With the boundary conditions   and 0 Solution of equation (1) and (2) yields in the cycle is given by in the cycle is given by Thus, the total cost of the system per unit time is given by The total profit   are considered as pentagonal fuzzy numbers, by using graded mean representation method for defuzzification, we have , the optimal value of 1 t and T can be obtained by solving the following equations:    2  1  1  01  0   2   2  2  1  5  1  5  05   2  2  1  4  1  4  04   2  2  1  3  1  3  03   2  2  1  2  1  02   2  2  1  1  1 Further, for the total cost function   The second derivative of the total cost function is complicated and it is very difficult to prove its convexity mathematically. Thus, the convexity of total cost function has been established graphically (figure 1).
Using Signed Distance Method the, total cost is given by The total cost function         Behavior of reliability, deterioration rate and holding cost in both the methods has been presented in the following figures.

VI. RESULT AND DISCUSSION
 It is indicated in Table 6 and Table 9 that as the value of r increases, total cost, total cycle length and time period of positive stock increase substantially.  It is indicated in Table 7 and Table 10 that as the value of 0  increases, total cost and total cycle length increase and time period of positive stock decreases gradually.  It is indicated in Table 8 and Table 11 that as the value of 0 h increases, total cost and total cycle length increase and time period of positive stock decreases drastically.

VII. CONCLUSION
In this paper, we have used signed distance method and graded mean representation method for defuzzying holding cost, setup cost, reliability, deterioration rate and shortage cost. These costs are taken as pentagonal fuzzy number. Demand is assumed to be an increasing function of time and reliability. It is seen from our analysis that increase reliability of the item gives rise to decrease cost as it is associated with quality assurances. So, the management has to fix up the reliability at certain level depending on the cost of investment.
The proposed model can be extended by introducing trade credit, weibull deterioration and partial backlogging.