1. Example A company purchases air filters at a rate of 800 per year $10 to place an order Unit cost is $25 per filter Inventory carry cost is $2/unit per year Shortage cost is $5 Lead time is 2 weeks Assume demand during lead time follows a uniform distribution from 0 to 200 Find (Q,R)

2. Solution Partial derivative outcomes:

3. Solution From Uniform U(0,200) distribution:

4. Solution Iteration 1: F(R) 2000 R

5. Solution Iteration 2:

6. Solution Iteration 3:

7. Solution R didn’t change => CONVERGENCE (Q*,R*) = (94,190) I(t) Slope -  253 159 190 With lead time equal to 2 weeks: SS = R –  =190-800(2/52)=159

8. Example Demand is Normally distributed with mean of 40 per week and a weekly variance of 8 The ordering cost is $50 Lead time is two weeks Shortages cost an estimated $5 per unit short to expedite orders to appease customers The holding cost is $0.0225 per week Find (Q,R)

9. Demand is per week. Lead time is two weeks long. Thus, during the lead time: Mean demand is 2(40) = 80 Variance is (2*8) = 16 Demand observed in one week is independent from demand observed in any other week: E(demand over 2 weeks) = E (2*demand over week 1) = 2 E(demand in a single week) = 2 μ = 80 Standard deviation over 2 weeks is σ = (2*8) 0.5 = 4 Solution

10. Finding Q and R, iteratively 1. Compute Q = EOQ. 2. Substitute Q in to Equation (2) and compute R. 3.Use R to compute average backorder level, n(R) to use in Equation (1). 4. Solve for Q using Equation (1). 5. Go to Step 2 until convergence.

11. Solution Iteration 1: From the standard normal table:

12. Solution Iteration 2: This is the unit normal loss expression. Table A - 4 gives values.

13. Solution Iteration 2: