# Question

The preceding problem describes the factors involved in making a managerial decision on the service level L to use. It also points out that for any given values of L, h (the unit holding cost per year), and σ (the standard deviation when the demand during the lead time has a normal distribution), the average annual holding cost for the safety stock would turn out to be C = hK1Lσ, where C denotes this holding cost and K1L is given in Appendix 5. Thus, the amount of variability in the demand, as measured by σ, has a major impact on this holding cost C.

The value of σ is substantially affected by the duration of the lead time. In particular, σ increases as the lead time increases. The purpose of this problem is to enable you to explore this relationship further.

To make this more concrete, suppose that the inventory system under consideration currently has the following values: L = 0.9, h =7 $100, and σ = 100 with a lead time of 4 days. However, the vendor being used to replenish inventory is proposing a change in the delivery schedule that would change your lead time. You want to determine how this would change σ and C.

We assume for this inventory system (as is commonly the case) that the demands on separate days are statistically independent. In this case, the relationship between σ and the lead time is given by the formula

σ = √dσ1,

where d = number of days in the lead time,

σ1 = standard deviation if d = 1.

(a) Calculate C for the current inventory system.

(b) Determine σ1 1. Then find how C would change if the lead time were reduced from 4 days to 1 day.

(c) How would C change if the lead time were doubled, from 4 days to 8 days?

(d) How long would the lead time need to be in order for C to double from its current value with a lead time of 4 days?

The value of σ is substantially affected by the duration of the lead time. In particular, σ increases as the lead time increases. The purpose of this problem is to enable you to explore this relationship further.

To make this more concrete, suppose that the inventory system under consideration currently has the following values: L = 0.9, h =7 $100, and σ = 100 with a lead time of 4 days. However, the vendor being used to replenish inventory is proposing a change in the delivery schedule that would change your lead time. You want to determine how this would change σ and C.

We assume for this inventory system (as is commonly the case) that the demands on separate days are statistically independent. In this case, the relationship between σ and the lead time is given by the formula

σ = √dσ1,

where d = number of days in the lead time,

σ1 = standard deviation if d = 1.

(a) Calculate C for the current inventory system.

(b) Determine σ1 1. Then find how C would change if the lead time were reduced from 4 days to 1 day.

(c) How would C change if the lead time were doubled, from 4 days to 8 days?

(d) How long would the lead time need to be in order for C to double from its current value with a lead time of 4 days?

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