Probabilistic Inventory Models
Probabilistic Inventory Models 1. CONTINUOUS REVIEW MODELS
1.1 "Probabilitized" EOQ Model
Some practitioners have sought to adapt the deterministic EOQ model to reflect the probabilistic nature of demand by using an approximation that superimposes a constant buffer stock on the inventory level throughout the entire planning horizon. The size of the buffer is determined such that the probability of running out of stock during lead time (the period between placing and receiving an order) does not exceed a prespecified value.
Let
L = Lead time between placing and receiving an order
[pic] = Random variable representing demand during lead time
[pic] = Average demand during lead time
[pic]= Standard deviation of demand during lead time
B = Buffer stock size a = Maximum allowable probability of running out of stock during lead time
The main assumption of the model is that the demand,[pic] ，during lead time L is normally distributed with mean [pic] and standard deviation [pic]-that is, N([pic], [pic])
FIGURE 14.1
Buffer stock imposed on the classical EOQ model
[pic]
Figure 14.1 depicts the relationship between the buffer stock, B, and the parameters of the deterministic EOQ model that include the lead time L, the average demand during lead time, [pic] , and the EOQ, y*. Note that L must equal the effective lead time.
The probability statement used to determine B can be written as
P {[pic] [pic] B +[pic]} [pic] [pic]
We can convert [pic] into a standard N (O, 1) random variable by using the following substitution
[pic]
Thus, we have
[pic]
Figure 14.2 defines [pic] such that
[pic]
Hence, the buffer size must satisfy
[pic]
The demand during the lead time L usually is described by a probability density function per unit time (e.g., per day or week), from which the distribution of the demand during L can be determined. Given that the demand per unit time is normal with mean D and standard deviation [pic], the mean and standard deviation, [pic] and
1.1 "Probabilitized" EOQ Model
Some practitioners have sought to adapt the deterministic EOQ model to reflect the probabilistic nature of demand by using an approximation that superimposes a constant buffer stock on the inventory level throughout the entire planning horizon. The size of the buffer is determined such that the probability of running out of stock during lead time (the period between placing and receiving an order) does not exceed a prespecified value.
Let
L = Lead time between placing and receiving an order
[pic] = Random variable representing demand during lead time
[pic] = Average demand during lead time
[pic]= Standard deviation of demand during lead time
B = Buffer stock size a = Maximum allowable probability of running out of stock during lead time
The main assumption of the model is that the demand,[pic] ，during lead time L is normally distributed with mean [pic] and standard deviation [pic]-that is, N([pic], [pic])
FIGURE 14.1
Buffer stock imposed on the classical EOQ model
[pic]
Figure 14.1 depicts the relationship between the buffer stock, B, and the parameters of the deterministic EOQ model that include the lead time L, the average demand during lead time, [pic] , and the EOQ, y*. Note that L must equal the effective lead time.
The probability statement used to determine B can be written as
P {[pic] [pic] B +[pic]} [pic] [pic]
We can convert [pic] into a standard N (O, 1) random variable by using the following substitution
[pic]
Thus, we have
[pic]
Figure 14.2 defines [pic] such that
[pic]
Hence, the buffer size must satisfy
[pic]
The demand during the lead time L usually is described by a probability density function per unit time (e.g., per day or week), from which the distribution of the demand during L can be determined. Given that the demand per unit time is normal with mean D and standard deviation [pic], the mean and standard deviation, [pic] and