Random demand

The stochastic tools used here differ considerably from those used in other fields of application, e.g., the investigation of measurements of physical data. For example, in this article normal distributions do not appear. On the other hand random sums, invented in actuary theory, are important. In the first theoretical part we start with random demand and end with conditional random service which is the basic quantity that should be used to decide how much of a product one should produce in a given period of time. 

Especially in the process industries various stochastic methods can be applied to cope with random demand. In many cases, random demands can be described by probability distributions, the parameters of which may be estimated from history. This is not always possible, the car industry is an example. No two cars are exactly the same and after a few years there is always a new model which may change the demand pattern significantly. 

Section 6.2 introduces the most important concepts in the description of random demand. Section 6.3 deals with random service, which results from random demand and production planning. Section 6.4 discusses the optimal planning of the production given specified random demands and known stock levels. Section 6.5 illustrates the theory by a few examples of the results applied to case studies. 

In this section we present the basic definitions and facts about random demand. In order to keep this paper self-contained we start with elementary definitions and facts. 

Three concepts are important random demand, random sums and conditional random demand. 

In applications it is often a matter of convenience if random demands are modeled by a continuous or by a discrete random variable. In both cases we require that a random demand can not be negative. In this section we assume continuous demands and assure the reader that all formulas have a straightforward extension to the discrete case. Therefore we usually skip the adjective continuous. 

The standard deviation can be used as a measure of risk. For example, if we pool the risk of random demands by several customers in one warehouse, the above formula indicates that the risk increases slower than the mean demand. It is therefore generally a good idea to pool risks. 

Gamma densities are often a suitable a priori choice to describe random demands, see, e.g., Tempelmeier [11], The reason for this is that they are not defined for negative numbers (demands are always positive) and that they are uniquely determined by their (positive) mean values and standard deviations. A further important property is that gamma densities are infinitely divisible, which means, for example, that there is a straightforward calculation of the daily demand from the weekly random demand. 

It is important to keep in mind that demands occur on the time line. If a reasonable minimal time interval is chosen, it is in many cases justified to consider the demands in these intervals as independent and identically distributed random variables. This means for example that the demand per week is the iterated convolution of the daily demand. A large customer base is a good indicator of independent random demand in different time intervals. 

If, for example, a product has independent daily random demand given by the gamma density with parameters X, c then the weekly demand is described by the gamma density with parameters X, 7c or X, Sc depending on the behavior of demand at weekends. In many cases gamma densities are a good approximation of random demand. If, however, for a product the amounts vary significantly from order to order, it may be necessary to consider the demand as a random sum. 

Compound densities, also called random sums, are typically applied in modeling random demands or random claim sums in actuarial theory. The reason for this is simple. Assume that customers randomly order different quantities of a product. Then the total quantity ordered is the random sum of a random number of orders. The conversion to the actuarial variant is obvious The total claim is made from the individual claims of a random number of damage events. Zero quantities usually are neither ordered nor claimed. This leads to the following definition. 

An order density is a demand density 5 with 5(0) = 0. The number of orders per interval can be described by a discrete density function t] with discrete probabilities defined for nonnegative integers 0,1, 2, 3, —The resulting (t], 8)-compounddensity Junction is constructed as follows A random number of random orders constitute the random demand. The random number of orders is r -distributed. The random orders are independent from the number of orders, and are independent and identically 5-distributed. 

Note that the mean /i (8) has a proportional influence on the standard deviation of the total ordered amount. The tendency is intuitive The formula means that only a few large orders require more safety stock to cover random demand variations than many small orders. It is therefore a good idea to measure the mean and the variance of demands twice. First in the usual way as mean and variance of a sequence of, say weekly, figures and then by analyzing the orders of a historic period and applying Eq. (6.6). The comparison of the two results obtained often provides insight in the independence and the randomness of the historic demand. If the deviation of the two mean values and/or two variances is large then the demand can not be considered as a random sum. A reason could be, for example, that the demand... 

In this overview article we only briefly discuss sporadic demand. We also do not deal with leaking demand or the like. However, the concepts and methods presented in the following sections for regular (i.e., plain) random demand can be extended to sporadic, seasonal and other sorts of demand. These extensions are natural but require much more difficult calculations. 

A random demand is not sporadic with respect to a period length if we can expect that the outcome for a period is almost surely greater than zero. In other words 8 (0) = 0 which does not mean that the outcome 0 is impossible if you throw a dice with infinitely many faces then any outcome has probability 0. 

In the first section we discussed random demand. Then we calculated the conditional demand and now finally we define conditional random service and conditional random shortage. These concepts are very important for optimization of service levels under capacity constraints. 

Random service levels result when random demands meet available inventories. Throughout this chapter we assume that the available inventory is not random but has a known value. This is justified because in many cases the production process is almost deterministic when compared with the varying demand. 

The mean service resulting from a random demand and an inventory is often divided by the mean demand and then called the service level. It is always within 0 and 1 and is usually written as a percentage value. Because the usage of this indicator implicitly assumes that partial deliveries are allowed, it is known as /S-service level in order to distinguish it from the a-service level, which denotes the proportion of completely serviced orders. 

First of all the resulting forecast is calculated, which is initially the conditional random demand resulting from the forecast and the orders that were already received, as discussed in Section 6.2.5. It is then balanced with the current inventory resulting in a forecast of the production demand. There may be periods when... 

Albritton, M., Shapiro, A., and Spearman, M. L. (1999), Finite Capacity Production Planning with Random Demand and Limited Information, Preprint. 

First, we describe numerically some tests carried out on the developed multiagent model, considering random demands, which follow certain statistical distributions. We have used samples with 30 and 90 temporary data. 

Table 1. Results of tests with random demands.

When analyzing the results, it is more appropriate to do it from a relative point of view that from an absolute one. When considering a larger number of data, and since the series in some cases have definite trends, the values of the Bullwhip Effect are significantly lower than in the cases analyzed with random demands. 

In this section, we consider a supply chain where a supplier sells to a retailer facing a random demand from customers. The target profit levels for the supplier and the retailer are set externally at and t, respectively. The retail price is fixed at r. The supplier procures or produces the good at a constant marginal cost c. Any unsold unit in the supply chain can be salvaged at a price v per unit. It is assumed v < c < r to avoid trivial situations. 

Parlar, M. 1988. Game theoretic analysis of the substitutable product inventory problem with random demands. Naval Research Logistics, Vol.35,397-409. 

X. Chen and D. Simchi-Levi. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost The finite horizon case. Working Paper, MIT, 2002a. 

L. J. Thomas. Price and production decisions with random demand. Operations Research, 22 513-518,1974. 

Eppen, G. and Schrage, L. (1981). Centralized Ordering Policies in a Multiwarehouse System with Random Lead Times and Random Demand . Management Science, 16, 51-67. 

Parlar, M., Game Theoretic Analysis Of The Substitutable Product Inventory Problem With Random Demands, Naval Research Logistics, 35 (1988), 397-409. 

Rosling, K. 1989. Optimal inventory policies for assembly systems under random demands. Operations Research. 37(4) 565-579. 

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