INVENTORY MODELING – Service Levels
· Welcome back.
· In this module we talk about determining the optimal reorder point when the distribution of demand during a fixed lead time is assumed to follow a probability distribution.
· Let’s refer to our example where the estimate for average yearly demand , D, was 6240.
· Now let’s assume that lead time is 8 working days,
· Then if we assume a company that operates 5 days per week, or 260 days per year
· And let’s assume a safety stock of 13 is desired.
· Then if demand was assumed to be constant during a lead time period of L years the optimal reorder point is r-star is L times D plus the safety stock.
· In this case L is 8 divided by 260 or .0308 years and D is 6240 per year,
· so if a safety stock of 13 were desired, the reorder point would be .0308 times 6240 plus 13 or 205.
· But even though on a long term basis, like a year, demand can appear relatively constant averaging 6240 per year,
· Suppose on a much shorter term, like the 8-day lead time in this problem, demand more closely follows a normal distribution with a certain mean and a certain standard deviation.
· Let us denote the average WEEKLY demand by mu sub W and
· The weekly variance by sigma-square sub W and the weekly standard deviation by sigma sub W.
· Then over any n-week period (where n need not be an integer value)
· We can model the total demand during this time period by a normal distribution with a mean of
· N times mu sub W and variance equal to
· N times sigma-squared sub W, meaning the standard deviation equals the square root of n times sigma sub W.
· Now over the course of a year, average yearly demand would equal 52 times mu sub W and the standard deviation would equal only the square root of 52 times sigma sub W.
· This standard deviation can be relatively small compared to the mean and hence assuming that over a long period (like 52 weeks) that average demand is relatively constant, is not a bad assumption. Hence we can simply used the basic EOQ formula developed earlier with D equal to 52 times mu sub W to calculate Q-star.
· But during a much shorter lead time, demand
· Is far less constant – in fact it can fluctuate quite a lot, hence we assume a normal distribution during lead time.
· Now to determine an appropriate safety stock and hence a reorder point, management must set what is called a target cycle service level for the inventory cycle – that it, it must designate an acceptable (usually small) probability, α, of running out of stock during an inventory cycle. 1 minus α would then be the probability that we did have enough inventory. This quantity, 1 minus α, is called the cycle service level.
· So suppose the length of a lead time period is L weeks and that estimates for the mean demand during a week mu sub W, and the standard deviation of demand during a week, sigma sub W, have been attained.
· Then the distribution of demand during this period is normal
· With a mean demand of L times mu sub W
· And standard deviation of the square root of L times sigma sub W
· Let’s return to our example for juicers sold by the Allen Appliance Company.
· Let’s assume that a normal distribution fairly accurately estimates demand over the course of a week.
· This can be checked using a chi-squared goodness of fit test or some other method.
· From our 10-week data, we can estimate the weekly demand to be the average demand of those last 10 weeks which was 120
· And we can estimate the variance in weekly demand by the sample variance over this 10 week period which turns out to be 83.33
· So that the standard deviation of weekly demand can be approximated by the square root of this value or 9.129
· So now we can model demand over an 8-day lead-time period by a
· Normal distribution
· With L = 8 days, it can be expressed in terms of weeks by 8 divided by 5 days per week or 1.6 weeks.
· Then the estimate for the mean demand during lead time is 1.6 times 120 or 192.
· And the estimate for the standard deviation of demand during lead time is the square root of 1.6 times the estimate of weekly standard deviation of 9.129 or 11.55.
· This leads us to the issue of safety stock.
· Suppose we wish to have a 99% cycle service level,
· that is, in the long run, we are willing to accept an average of 1 stockout in every 100 inventory cycles.
· Now the distribution of demand during this lead time is normal
· With a mean of 192
· And a standard deviation of 11.55
· And we want the x value that puts probability .99 to its left, that is probability .01 in the upper tail
· On the z-scale
· This is z sub .01, which we can look up as 2.33
· Thus the required reorder point is 2.33 standard deviations above the mean value of 192, that is 192 plus 2.33 times 11.55 which is approximately 219
· Thus the reorder point is 219 and the safety stock is then the difference between the reorder point and the average demand during the period, 219 minus 192, or 27.
· So r-star is found by
· Mu sub L plus z sub .01 times sigma sub L or
· The safety stock is just the “plus part” of this last expression – z sub .01 times sigma sub L, which is about 27
· The corresponding annual safety stock cost is then generated by C sub H times this safety stock or $1.40 times 27 which is $37.80
· This total annual safety stock cost should be a component of the total annual cost of the policy.
· We can find the reorder point and calculate the safety stock
· By using the Cycle Service Level worksheet of the inventory template.
· We enter the mean and standard deviation of demand during the lead time period and the cycle service level
· This generates the reorder point.
· The safety stock can then be calculated by this reorder point minus the mean value (in cell B5), which can then be entered on the EOQ template.
· Let’s review what we’ve discussed in this module.
· We’ve stated while long term demand might well be modeled as being relatively constant, short term demand can be modeled by a probability distribution such as a normal distribution.
· Then we would use the assumption that long term demand is relatively constant to use the EOQ formula to calculate the optimal order quantity, Q-star.
· But given an assigned cycle service level of 1 minus α
· We can determine a mean, mu sub L and a standard deviation, sigma sub L, of demand during a lead-time period.
· The safety stock is found by z sub α times sigma sub L and the reorder point is found by adding this safety stock to the mean demand for the interval, mu sub L.
· The corresponding total annual safety stock costs are found by C sub H times the safety stock. These total annual safety stock costs should be accounted for in the total annual cost calculation.
· Finally we showed how to use the Cycle Service Level worksheet of the inventory template to calculate the reorder point.
That’s it for this module. Do any assigned homework and I’ll be back to talk to you again next time.