In Miltenburg’s coordinated control system, the residual stock of a particular item is denoted by the symbol z, and in the case of periodic review, it turns out to be reasonable to approximate its distribution by a normal distribution with mean μz and standard deviation σz .

a. Why is there a spike at z = 0 in the continuous review case but not in the periodic review case? Also, why can z be negative in the periodic review situation?

b. Using the following equation and assuming that the demand x during R+L is normally distributed with mean xˆR+L and standard deviation σR+L, find as simple an expression as possible that the reorder point s must satisfy to ensure a probability of no stockout equal to P1.
Pr{Stockout} = Pr{x ≥ s + z}
= ∞
z0=−∞
fz (z0)dz0 ∞
x=s+z fx(x0)dx0 Hint: Use the following result proved by Miltenburg (1982).
∞
−∞
1 √2πσz exp[−(z0 − μz )
2/2σ2 z ]
pu≥
s + z0 − ˆxR+L σR+L 
dz0 = pu≥
s + μz − ˆxR+L cσR+L 
where c = 
1 + σ2 z /σ2 R+L

c. Find s for the case where P1 = 0.96, μz = 5.0, σz = 2.3, xˆR+L = 12.0, and σR+L = 4.2. What s value would be used if the residual stock was ignored? P-96