Stochastic Demand

Overview


Stochastic Demand


One way to extend the inventory models is to assume that the demand for purchasing product is random. That is, the firm cannot assume that the demand per period is constant. This means that the firm will not know ahead of time how much inventory to purchase in order to cover demand over the next period.

One common way to model demand is to assume that it is normally distributed. (see normal distribution) The normal distribution is as standard distribution for many random variables, primarily because of the law of large numbers. In the case of inventory demand, it is a reasonable assumption, but not perfect. The normal distribution will assign positive probabilities to negative demand, and will assign postive (although very small) probability to demand that is arbitrarily large.

If the unbounded nature of the normal distribution is unacceptable, you could choose to use the beta distribution, which is bounded both above and below.

Once you assume that demand is normal, you need to specify the average dmand and the standard deviation. Once that is done, the distribution is fixed and can be computed. You can compute the probability that demand will exceed inventory

{% \alpha = F_N (inventory, \mu, \sigma) %}

where {% \alpha %} is the probability, {% \mu %} is the average demand and {% \sigma %} is the standard deviation.

Likewise, one can compute the amount of inventory to purchase to cover the next periods demand within a given probability. (i.e. how much inventory to purchase so that there is only a 5% chance of running out over the period.)

{% purchase = F_N^{-1} (\alpha, \mu, \sigma) %}

Try it!

Remembering the summation formula for normal distributions
{% X \sim N (\mu_x, \sigma_x^2) %}
{% Y \sim N (\mu_y, \sigma_y^2) %}
{% Z = X+Y %}
{% Z \sim N (\mu_x +\mu_y, \sigma_x^2 + \sigma_y^2) %}
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One can compute the demand distribution for any number of periods.

Correlated Demand

The above formula assumes that demand from one period to the next are indpendent (or uncorrelated). In practice, this may be a bad assumption. However, this is probably unrealistic. A simple correction to the variance of the normal distribution inlcudes the correlation

{% \sigma_{x+y}^2 = \sigma_x^2 + \sigma_y^2 + 2\rho \sigma_x \sigma_y %}

For futher information on how to fit distributions to data, please see

fitting distributions

Stochastic Lead Time


In addition to modeling the demand per period stochastically, we can also model the lead time (time between placing an order and receiving the order) stochastically. Generally, it is assumed that lead times are distributed normally. (with the same caveats listed above for modeling the demand.)

Now the question is how to get a distribution that can be used to derive the relevant probabilities, i.e. what is the probability that we will run out of inventory after we place an order for more. In this case, we dont know exactly how long the order fulfillment time will be after we place an order, and we dont know how much demand will occur during that period. However, we have distribution functions for each variable.

One might think that the distribution is the distribution that results from multiplying the two random variables, i.e.

{% Amount \: of \: Demand \: during \: FulFillment = Lead \: Time \times Demand \: per \: Period %}

This is close but not exact. Well think about it from a slightly different perspective. Assume that we are measuring the demand per period as demand per day, and assume that the average lead times are several days. That is, we use a period for the demand per period that is smaller than the lead time. Lets suppose that the lead time turns out to be 10 days. (That is, we have chosen a lead period from the lead time distribution and it has turned out to be 10 days). Then the amount of demand will be
{% demand = \sum_1^{10} Normal %}


That is, it is the sum of 10 draws from the normal demand distribution. This is a sum of a random number of variables. In this case, this will simplify the problem, because we know that the sum of normal variables is normal, so all we need is the mean and variance. In this case, the mean and variance is given by

{% mean = mean_{lead \: time} \times mean_{demand}%}


{% variance = mean_{lead \: time} \times variance_{demand} + variance_{lead\:time}\times mean_{demand}^2%}

reference

Now that we know the distribution is normal and we have the mean and variance, we have the desired distribution.

Of course, coming up with an exact distribution is not necessarily a critical issue, because the distributions we choose for the underlying random variables is only approximate, that is, we know that they are not normal, so we could be forgiven for not using the above logic and using a different distribution instead.

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