Example

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. 0 200 R Solution • Iteration 1: F(R)

5. Solution • Iteration 2:

6. Solution • Iteration 3:

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

10. Finding Q and R, iteratively • 1. Compute Q = EOQ. • 2. Substitute Q in to Equation (2) and compute R. • 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: