1. INVENTORY MODELING Service Levels

2. DETERMINING A REORDER POINT, r* (With Safety Stock) Suppose lead time is 8 working days The company operates 260 days per year Suppose a safety stock of SS = 13 is desired L = 8/260 .0308 yrs D = 6240 /year r*205 r* =.0308(6240) +13  192 +13 = 205 Reorder Point r* = LD + SS Where L and D are in same time units

3. Actual Demand Distribution normal distributionSuppose on a short term basis demand actually more closely follows a normal distribution with:  W –Weekly mean demand =  W  2 W  W –Weekly variance =  2 W, Weekly St’d dev. =  W Demand over an n-week period: –normal n  W –Mean n  W n  2 W (  n)  W –Variance = n  2 W, St’d Dev. = (  n)  W _

4. Calculating Q* Over the course of a year, the standard deviation becomes small relative to the mean value -- hence a common practice is to ignore any variability and calculate Q* by the usual EOQ formula

5. Lead Time Demand Lead times, however, tend to be short and hence variability must be considered. cycle service level the probability of not running out of stock during the lead time periodA cycle service level is supplied to the modeler -- the probability of not running out of stock during the lead time period. LweeksSuppose lead time is L weeks and estimates for μ W and σ W have been attained. –Demand during lead time is normal L  W –Mean demand =  L = L  W  L  W –St’d dev. =  L =  L  W _

6. Example -- Allen Appliance Suppose we can assume that demand follows a normal distribution –This can be checked by a “goodness of fit” test From our data, over the course of a week, W, we can approximate  W by (105 + … + 130)/10 = 120  W 2  s W 2 = ((105 2 +…+130 2 ) - 10(120) 2 )/9  83.33 Then σ W  =  83.33 = 9.129

7. DEMAND DISTRIBUTION DURING 8 -DAY LEAD TIME Normal distribution for lead time demand L = 8 days = 8/5 = 1.6 weeks, so  L 192  L = (1.6)(120) = 192  L 11.55  L   1.6 (9.129) = 11.55

8. XZXZ SAFETY STOCK Suppose we wish a cycle service level of 99% –WE wish NOT to run out of stock in 99% of our inventory cycles 0Z.01 = 2.33.01  L = 11.55 192 ? 192 + 2.33(11.55) 219 R* = 219 SS = 219 – 192 = 27

9. Calculating r* and Safety Stock Costs r*Reorder point, r* =  L + z.01  L = 219 192 + 2.33(11.55)  219 SS27Safety stock SS = 2.33(11.55) = 27 Safety stock cost$37.80Safety stock cost = C h SS = 1.40(27) = $37.80 This should be added to the TOTAL ANNUAL COST

10. Using the Template Enter Lead Time Information Select Cycle Service Level Worksheet Reorder Point Round r* to 219 Enter SS = 219 – 192 = 27 Into SS of appropriate worksheet

11. Review In the short run, demand may seem to follow a probability distribution (normal) In the long term, variability is relatively insignificant in magnitude compared to the mean value-- so calculate Q* in usual way. 1- Determine a cycle service level = 1-  Determine the mean and st’d deviation for demand during lead time L r* =  L + SSSS = z   L r* =  L + SS C h SSSafety Stock Costs = C h SS - add to total cost Use of Template