Overview

This paper covers a methodology to ensure parts availability and optimize carrying and acquisition costs by combining equipment criticality, stock item usage and lead-time data, and computerized data analysis into an automated system for calculating inventory levels.

Statistical Analysis of Data

The basis for analysis of lead-time and inventory issues is that these variables (represented by X₁…X_n) are independent random samples such that the distribution is normal and can be represented by Equation 1.¹  Data analysis for either lead-time of inventory issues is assumed valid when:

1. There are three or more data points for each

2.  The standard deviation is less than the mean for each

Inventory levels should be calculated manually reviewed when these conditions are not met.

Equation 1       m ± zs/sqrt (n)  Where: m          =          Sample mean or average z          =          Factor based on the degree of confidence that any value Xi will fall within the boundaries of Equation 1 s          =          Standard deviation = SQRT ((S(Xi) 2 – (S(Xi)) 2/N)/(N-1)) n          =          Number of values Xi

Inventory management deals only with maximum values of lead-time and usage and therefore variance from the mean is only evaluated on one side.  We have no concern if a lead-time is shorter than expected, or a part is not needed when its use is expected.  Such an evaluation is called a one-sided or one-tailed evaluation and has specific z factors that differ from those used for a two-sided evaluation.

Confidence Levels

Confidence levels give the probability that any value of X_i will fall within the boundaries defined by Equation 1.  Confidence levels are arbitrary values, which for inventory purposes are assigned based on the criticality of equipment or commodity type (see Table 1).  The equipment criticality category is determined by evaluation of equipment importance with respect to process safety and capability (A is most critical and D least critical).²  Criticality values for inventory items are generated using the procedure outlined in Reference 2.   Default values of  “B” for equipment repair parts and “C” for general issue items can be used where criticality values are not available.

Lead-time

Lead-time (L) is the time period from when the stock level reorder point is reached to the time that new materials are received in the warehouse.   This includes the time between stock item issue and generation of a purchase requisition.  

Usage

Usage data is calculated once the lead-time interval has been determined.  Data points are grouped as the number of stock issues in the lead-time interval L_cl, with N_i …N_n subsets in the total stock issue date range such that Equation 3 applies.  Each subset N_i contains U_N stock item issues.   Treatment of usage data as such ensures that factors such as infant mortality of equipment are captured.  The calculation for usage should capture only warehouse issues associated only with unplanned maintenance.  Project and planned maintenance inventory usages are based on random events and therefore should be treated separately.

Economic Order Quantity

The economic order quantity (EOQ) considers carrying costs, acquisition costs, and usage in the calculation for optimum order amounts.  Equation 5 gives the EOQ formula.  Values for carrying cost and acquisition cost (as fraction of purchase order cost) are used as constants in this formula and should be verified and documented.   

Minimum Inventory Levels

The minimum inventory level (Min) for a given stock item is the maximum of the following three values:

1. Minimum issue quantity: the minimum or typical amount used for a repair.

2. Arbitrary percent of EOQ: this applies for low-cost items.  For example, there is no sense running out of tab washers on a million-dollar power generation unit when they cost only five cents each.   The default calculation for percent EOQ can be set at 10 percent.  An upper boundary such as percentage of Condition 1 or 3 may be required.

3. Maximum usage minus average usage: this difference is calculated over the lead-time interval L_M as [U_CLL_CL-U_ML_M].

Maximum Inventory Levels

The maximum inventory level (Max) for a given stock item is the minimum of the following three values but not less than the reorder point:

1. Sum of the minimum inventory level plus the EOQ

2. Maximum usage in given time period calculated as [(average usage) * (time period)].  The initial default for time period can be set at two years.  

3. Maximum usage in the shelf life of inventory item calculated as [(average usage) * (shelf life) ] 

Reorder Point

The reorder point (RP) for a given stock item is the maximum of the following two values:

1. Maximum usage over lead-time interval  (U_CLL_CL )

2. Minimum inventory level plus one

Reorder Quantity

The reorder quantity is the difference between the maximum and minimum stock item quantities.

Data Trends

Changes in equipment/component population, reliability, and other factors such as seasonal usage impact inventory levels and should be captured.  Since usage can be equated to population divided by reliability as shown in Equation 6, a usage value U_C can be calculated as a direct ratio of population or reliability change.  This calculated value, U_c, is a user-entered value substituted for UCL in all inventory calculations for a period of MTBF_C.   Seasonal usage calculations and adjustments are outside the scope of this paper.
 

Equation 6       UC     =          PC/MTBFC             Where:              UC         =        Usage of a component  Annual Usage (average)             PC         =        Population of a component                        MTBFC   =       Mean time between failures for a component

Data Validity Testing and Exception Reporting

Experienced personnel should review calculated inventory control values for validity.  Exception reports can be used to identify changes that occur to critical items or changes that exceed an arbitrary percentage over specified time periods.  For example, an exception report might identify any items of criticality “A” with changes that exceed 25 and 50 percent of the original values.

References:

1.        Bhattacharyya, Gouri K and Johnson, Richard A., 1940, Statistical Concepts and Methods, John Wiley and Sons, Inc., 1977.

2.        Ciliberti, V. Anthony, Use Criticality-Based Maintenance for Optimum Equipment Reliability, Chemical Engineering Progress, July 1998.