If the random component is small compared to the deterministic component, the models of chapter 4 will be accurate. If not, randomness must be explicitly accounted for in the model.
In this chapter, assume that demand is a random variable with cumulative probability distribution F(t) and probability density function f(t).
These models have the objective of properly balancing the cost of Underage – having not ordered enough products vs. Overage – having ordered more than we can sell
These models apply to problems like:
Planning initial shipments of ‘High-Fashion’ items
Amount of perishable food products
Item with short shelf life (like the daily newspaper)
Because of this last problem type, this class of problems is typically called the “Newsboy” problem
At the start of each day, a newsboy must decide on the number of papers to purchase. Daily sales cannot be predicted exactly , and are represented by the random variable, D.
The newsboy must carefully consider these costs: c o = unit cost of overage
c u = unit cost of underage
It can be shown that the optimal number of papers to purchase is the fractile of the demand distribution given by F(Q*) = c u / (c u + c o ).
6.
Determination of the Optimal Order Quantity for Newsboy Example
Earlier we considered reorder points (number of parts on hand when we placed an order) they were dependent on lead times as a dependent variable on Q, now we will consider R as an independent variable just like Q
Assumptions:
Inventory levels are reviewed continuously (the level of on-hand inventory is known at all times)
Demand is random but the mean and variance of demand are constant. (stationary demand)
The response time of the system (in this case) is the time that elapses from the point an order is placed until it arrives. Hence,
The uncertainty that must be protected against is the uncertainty of demand during the lead time.
We assume that D represents the demand during the lead time and has probability distribution F(t). Although the theory applies to any form of F(t), we assume that it follows a normal distribution for calculation purposes.
For the basic EOQ model discussed in Chapter 4, there was only the single decision variable Q .
The value of the reorder level, R , was determined by Q.
Now we treat Q and R as independent decision variables.
Essentially, R is chosen to protect against uncertainty of demand during the lead time, and Q is chosen to balance the holding and set-up costs. (Refer to Figure 5-5)
18.
Changes in Inventory Over Time for Continuous-Review (Q, R) System
Interpret n(R) as the expected number of stockouts per cycle given by the loss integral formula (see Table A-4 (std. values)). And note, the last term is this cost model is a shortage cost term
The optimal values of (Q,R) that minimizes G(Q,R) can be shown to be:
In many circumstances, the penalty cost, p , is difficult to estimate. For this reason, it is common business practice to set inventory levels to meet a specified service objective instead. The two most common service objectives are:
Type 1 service: Choose R so that the probability of not stocking out in the lead time is equal to a specified value.
Type 2 service. Choose both Q and R so that the proportion of demands satisfied from stock equals a specified value.
For type 1 service, if the desired service level is α then one finds R from F(R)= α and Q=EOQ.
Type 2 service requires a complex interative solution procedure to find the best Q and R . However, setting Q=EOQ and finding R to satisfy n(R) = (1-β)Q (which requires Table A-4) will generally give good results.
For a type 1 service objective there are two cycles out of ten in which a stockout occurs, so the type 1 service level is 80%. For type 2 service, there are a total of 1,450 units demand and 55 stockouts (which means that 1,395 demand are satisfied). This translates to a 96% fill rate.
The (Q,R) policy is appropriate when inventory levels are reviewed continuously. In the case of periodic review, a slight alteration of this policy is required. Define two levels, s < S, and let u be the starting inventory at the beginning of a period. Then
(In general, computing the optimal values of s and S is much more difficult than computing Q and R.)
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