توجه بر روی تحقیق "کوجا و پارک در نشریه امگا 31 (2003)،با موضوع: تعیین اندازه دسته تولید بهینه تحت کاهش قیمت مداوم"

Khouja and Park [1] analyze the problem of optimizing the lot size under continuous price decrease. They show that the classic EOQ formula can lead to far from optimal solutions and develop an alternative lot size formula using the software package Mathematica. This formula is more exact, but also more complicated. In this note, we study the net present value formulation of the model, and thereby gain an insight that leads to the proposal of a modified EOQ formula. The modified EOQ formula, albeit not as accurate, is a good alternative to the formula developed by Khouja and Park, especially if mathematical complexity may hamper implementation.

Khouja and Park [1] analyze the problem of optimizing the lot size under continuous price decrease. This problem is relevant for the high-tech industry and especially the PC assembly industry, where the prices of components decrease at significant rates. They study a single-item model with a constant lead time, constant demand, no quantity discounts, and no shortages allowed. However, their model deviates from the standard economic order quantity (EOQ) model in two ways: (i) there is a finite planning horizon, and (ii) the purchase price decreases at a constant rate. Khouja and Park develop an expression for the total cost over the planning horizon using a mixture of the average cost (AC) approach and the net present value (NPV) approach. They continuously discount the price as in an NPV approach, but charge an interest cost per time unit rather than discount purchase cost. By setting the derivative of the cost expression to zero and using a Taylor series approximation for one of the exponential terms, they derive a complex optimality condition for the number of orders during the planning horizon. Using the software package Mathematica, they then find an expression for the number of orders during the planning horizon, which leads to nearly optimal solutions for realistic values of the model parameters. They also develop the corresponding expression for a nearly optimal order quantity. For a specific example, Khouja and Park illustrate that their order quantity formula indeed leads to a nearly optimal solution. They further show for this example, that the classic EOQ formula, with holding cost per unit of inventory value per time unit equal to the interest rate, results in a far from optimal solution. As mentioned above, Khouja and Park use a mixture of the AC approach and the NPV approach in deriving their total cost expression. In this note, we instead develop a `pure' NPV expression. Although the numerical difference between the expressions is small for examples with realistic parameter settings, the pure NPV expression leads to the important insight that the holding cost per unit of inventory value per time unit in the `corresponding' AC approximation is equal to the interest rate plus the rate of price decrease. We therefore propose a modified version of the classic EOQ formula with holding cost per unit of inventory value per time unit equal to the interest rate plus the rate of price decrease. We illustrate for the example of Khouja and Park, that the modified EOQ formula leads to a nearly optimal solution. An extensive numerical experiment shows that this result also holds in general. Combining this near-optimality with the simple structure of the EOQ formula that many practitioners are familiar with, we conclude that the modified EOQ formula has great practical value. The remainder of this paper is organized as follows. In Sections 2 and 3, we review the model and the results of Khouja and Park [1]. In Section 4 we apply the pure NPV approach and present our results. We end with conclusions in Section 5.

An NPV formulation of Khouja and Park's model with a continuously decreasing price, lead to the insight that the holding cost per unit of inventory value per time unit in the corresponding AC model should be equal to the sum of the interest rate and the rate of price decrease. This lead to the proposal of a modified EOQ formula. In an extensive numerical experiment, it was shown that the modified EOQ is a good approximation of the optimal order quantity. The average increase in the NPV compared to the optimal solution was only 0.03%, which is much lower than the 0.98% for the classic EOQ. The order quantity formula proposed by Khouja and Park is even more accurate. In the same experiment, it gives an average increase in the NPV of 0.001%. However, that formula is rather complex, which may hamper its implementation in situations where users have limited mathematical skills. In such situation, the modified EOQ formula is a good alternative, especially if users are already familiar with the classic EOQ formula.