Purchasing Inventory Models for Deteriorating Items with Quadratic Demand

This paper deals with purchasing inventory replenishment policy for deteriorating items consider with the time-dependent quadratic demand and time-dependent backlogging. Two models were formulated and solved. First, it is for deteriorating items with quadratically time-dependent demand for deteriorating items. Second, quadratically time-dependent demand for deteriorating items and shortages. A mathematical model is developed to the fourth-order equation for each model, and the optimal production lot size, which minimizes the total cost is derived. Sensitivity analysis is carried out to demonstrate the effects of changing parameter values on the optimal solution of the system. Numerical examples are taken to illustrate the procedure of finding the optimal inventory cost, cycle time, and optimal lot size. The numerical experiment in this model was coded in Microsoft Visual Basic 6.0. Lot


Introduction
The Classical inventory models assume constant demand in an endless planning horizon [1]. This assumption is valid during the maturation process of the product life cycle and the finite span. The inventory can be improved in other stages of the product life cycle. It is a positive business launch or a decline due to the introduction of new rival goods. However, demand is not always constant in reality and the market is called by time-dependent. The main issue of the current paper and inventory model is the timedependent quadratic demand, time-proportional deterioration, and backlog. Most inventory modelers consider time-varying demand. It is known to be linear and exponential in two types of time-dependent demands. Linear time-dependent demand D = a + bt implies a steady increase or decrease in demand according as b > 0 (or) b < 0. It is unrealistic in the real market. On the other hand, an exponentially time-varying demand D = De at is unrealistic because, in real market situations, demand is unlikely to vary a high rate as exponential. Therefore, the time-dependent quadratic demand of the form = + + 2 D a bt ct , a>0, seems to be a better representation of time-varying market demand.

Assumptions
In this paper, we use assumption is used to formulate the problem such as: (1) Initially, the inventory level is zero; (2) The demand rate is a quadratic function = + + 2 D a bt ct where a>0,  0, 0, bcat time t and it is a continuous function of time.
Here, a, b, and c are constants and "a" stands for the initial demand "b" and "c" are a positive trend in demand; (3) The holding cost, ordering cost and shortage cost remain constant over time; (4) The planning horizon is finite; (5) The lead time is zero; and (6) There is no repair or replacement of the deteriorated items.

Notations
Notations are used in this paper such as;

Purchasing Inventory Model for Deteriorating Items with Quadratic Demand
The typical behavior of the inventory model has depicted in Fig. 1. The inventory starts with zero stock at zero time. In most of the inventory models, demand has considered a constant function. Nevertheless, in a realistic situation, demand is always not a constant function. It is varying according to time. Therefore, this model developed a deterministic deteriorating inventory model. In which the demand is a quadratic function of time, that is at time t and "a" stands for the initial demand, and "b" is a positive trend in demand, and the deterioration rate is constant.
The following differential equation gives the instantaneous inventory level I(t). with the boundary conditions are (0) = ; ( ) = 0 The solution of equation (1) is shown in equation (3).
The total cost includes the sum of the cost of installation, the cost of holding, and the cost of deterioration. Setup cost, holding cost, deteriorating cost, and The total cost respectively are shown in equation (4), (5), (6), and (7).
To solve TC optimal, it can be easily shown that TC (T) is a convex function in T. Hence, an optimal cycle time T can be calculated from: Equation (8) is the equation for the optimum solution for the total cost. It should be noted that, When b =0 , c=0 and a = D, then which is the standard model.

Purchasing Inventory Model for Deteriorating Items with Quadratic Demand and Shortage
The typical behavior of the inventory model is depicted in Fig. 2. The inventory starts with zero stock at zero time. Shortage at 1 T is to accumulate at the early stage of the inventory cycle. Therefore, this model developed the deteriorating inventory model. The demand is a linear function of time. That is at time t and "a" stands for the initial demand and "b" and "c" are a positive trend in demand. Therefore, the deterioration rate is constant.
The instantaneous shortage I(t) during the period of shortage is given by the derivative of equation (10).
with the boundary conditions are The solution of equation (9) is The solution of the equation (10) is From equation (9) and boundary conditions Total cost comprised the sum of the setup cost, holding cost, deteriorating cost, shortage cost. Setup cost, holding cost, deteriorating cost, Shortage cost and The total cost respectively are shown in equation (14), (15), (16), (17) and (18).  1   2  2  3  1  1  1  1  1  1  2  3   2  2  1  1  1  1  1  1  2  2  3  2  3   2  1  2  2  3   2  2  1  2   h   T   a bT  cT  b  cT  aT  bT  cT  c   C  T  bT  cT  cT  a bT  cT  b cT c e The  is discarded since it is minimal, so.   T  T  T  c  T  TT  b  T  T  a  T   C   T   cT  bT  aT  C  C   TC  T Partially derivative of the total cost equation (18) to T is It should be noted that when b =0 and a = D, then ) ( Which is the standard model.

Numerical Experiment
In the model for deteriorating items with quadratic demand, we used D = 4500 unit, In the sensitivity test of the Rate of deteriorating items toward the total cost, this study used ten variations of the deteriorating rate from 0.01 to 0.10, and both models performed this test. Sensitivity analysis was also carried out on parameters such as ordering cost, holding cost, deterioration cost, shortage cost, initial demand (a), and positive trend (b,c). During the experiment, 5 parameter variations were used to determine its effect on inventory. To simplify the experiment, numerical experiments in this model was coded in Microsoft Visual Basic 6.0.

Model for Deteriorating Items with Quadratic Demand
Result from numerical experiment show that the optimal solution is T = 0.0618, Q* = 278.10, Setup cost = 1618.69 $, Holding cost = 1390.54 $, Deteriorating cost = 139.05 $, and Total cost = 3147.70 $. Furthermore, Table 1 shows the Influence variation of rate of deteriorating items toward the total cost. It is concluded that The greater the rate of deteriorating, it increases the Setup cost, Deteriorating cost, and Total cost. Conversely, the greater the rate of deteriorating, the lowering T, Q, and holding cost.  Table 2 shows the effect of demand and cost parameters on optimal values toward T, Q, setup cost, holding cost, deteriorating cost, shortage cost, and total cost. The following influences can be obtained from sensitivity analysis based on Table 2. The finding of sensitivity analysis is as follows 1) Increasing the setup and deteriorating cost raise optimum quantity, cycle time, setup cost, holding cost, deteriorating cost, and total cost increases. These results show that there is a positive relationship between them. 2) Increasing the holding cost also raise the setup cost, holding cost, Deteriorating cost, and total cost. There is a positive relationship between them, but the optimal cycle time and the optimal lot size decreases. Therefore, there is a negative relationship between cycle time and an optimal lot size toward holding costs. Demand parameters a, b, and c also affect the optimal quantity, cycle time, setup cost, holding cost, deteriorating cost, and total cost. When a, b, and c increase, it will increase the setup cost, holding cost, deteriorating cost, and total cost. However, the optimal quantity, cycle time decreases.

Model for Deteriorating Items with Quadratic Demand and Shortage
In the trial model for deteriorating items with quadratic demand and shortage, the results show  Table 3. The study considers the effect of the rate of deteriorating items toward cycle time, optimum quantity, set-up costs, holding costs, deteriorating costs, shortage cost, and total costs. It is concluded that increasing the rate of deterioration of the item enhances the set-up cost, the deterioration costs, shortage cost, and the total cost. These show that there is a positive relationship between them. However, when the rate of deterioration increases, then the cycle time, the optimum quantity, and the holding cost decreases. It shows that there is a negative relationship between them.  The effect of Demand and cost parameters on optimal values can be shown in Table 4. The experimental findings are as follows 1) increasing in setup cost per unit can raise optimum quantity, cycle time, setup cost, holding cost, deteriorating cost, shortage cost, and total cost increases. Therefore, there is a positive relationship between them. 2) increasing in holding cost per unit time enhance the setup cost, holding cost, shortage cost, and total cost. There is a positive relationship between them, but optimal cycle time, optimal lot size, and deteriorating cost decrease. Therefore, there is a negative relationship between holding cost per unit time toward cycle time, lot size, and deteriorating cost. 3) increasing deteriorating cost per unit time enhance the setup cost, deteriorating cost, shortage cost, and total cost. However, it decreases cycle time, lot size, and holding cost. 4) Raising in the shortage cost decreases the optimal cycle time, optimal lot size, and shortage cost. However, the replacement time, maximum inventory, setup cost, holding cost, and total cost show increase. Other parameters "a", "b", and "c" can be observed from Table 4.

Conclusion
This paper deals with purchasing inventory replenishment policy for deteriorating items. This model, considering the time-dependent quadratic demand. Two models were formulated and solved for deteriorating items with (1) quadratically time-dependent demand for deteriorating items, and (2) quadratically time-dependent demand for deteriorating items and shortages. The sensitivity analysis also was presented in both models. The proposed model can assist the manufacturer and retailer in the accurate determination of the optimum quantity, cycle time, and overall inventory cost. This research can be extended as follows: (1) Most of the production systems today are multi-stage systems, and in a multi-stage system, the defective items and scrap can be produced in each stage. The defectives and scrap for a multi-stage system can be different stages. (2) Inspection costs can be included in developing future models. (3) The demand for a product may decrease with time owing to the introduction of a new product, which is either technically superior or more attractive and cheaper than the old one. On the other hand, the demand for new products will increase. Thus, the demand rate can be varied with time, so the variable demand rate can be used to develop the model.