Single-Echelon Systems: Reorder Points

Abstract

We shall now consider different techniques for determining reorder points, or equivalently safety stocks, when the demand is a stationary stochastic process. To do this we first of all need a suitable demand model. In practice the demand during a certain time is nearly always a nonnegative integer, i.e., it is a discrete stochastic variable. (Exceptions may occur when we deal with products like oil.) Provided that the demand is reasonably low, it is then natural to use a discrete demand model, which resembles the real demand. However, if the demand is relatively large, it may be more practical to use a continuous demand model as an approximation. See Sect. 5.2. As before it is assumed that individual items can be controlled separately.

Problems

(*Answer and/or hint in Appendix 1)

• 5.1* In Sect. 5.1.1 it is shown how a compound Poisson process with f ₀ > 0 can be replaced by an equivalent process with f ₀´ = 0. Prove this.

• 5.2* Consider compound Poisson demand with a logarithmic compounding distribution as in Sect. 5.1.2. Show that D(t) has a negative binomial distribution.

• 5.3 5.3 Let μ´ = 3 and σ´ = 1. Make a computer program that fits a distribution of D(t) to these values according to Sect. 5.1.4. Verify from the obtained distribution that you get the correct μ´ and σ´.

• 5.4 Show that G(x) → − x as x → − ∞. Use this to show that F(x) in (5.42) app- roaches 1 as x → ∞.

• 5.5 Consider the density of the inventory level f(x) in (5.43). Show that f(x) has its maximum at x = R − μ´ + Q/2.

• 5.6* The lead-time demand for an item is normally distributed with mean 300 and standard deviation 100.

  • (a) Set S ₁ to 75, 85, and 95 %. Determine the corresponding safety stocks.

  • (b) Set the safety factor to 1. Determine S ₂ for Q = 100, 500, and 1000.

  • (c) Assume that the average demand per week is 25 and 100, respectively. Let the safety stock be determined as the average demand during three weeks. Determine the corresponding values of S ₁.

• 5.7* A company plans to use an (R, Q) policy with continuous review for inventtory control. A certain test item is considered. Demand is regarded to be continuous. The lead-time demand has mean 100 and standard deviation 50. The policy R = 135 and Q = 200 is considered.

  • (a) Determine S ₁ and S ₂ for normally distributed lead-time demand.

  • (b) Assume instead that the lead-time demand has a uniform distribution with the same mean and standard deviation. Determine S ₁ and S ₂.

• 5.8* A company tries to reduce all lead-times by 50 %. There is a discussion concerning the impact of this change. As an example an item in stock is considered. Demand is continuous and normally distributed. Inventory control is carried out by an (R, Q) policy with continuous review. The demand per week has mean 200 and standard deviation 50. Demands during different weeks are independent. From the beginning the lead-time is four weeks and R = 850 and Q = 600. Two questions have been raised:

  • (a) Assume that the lead-time is reduced to two weeks, while R and Q are unchanged. How much does S ₂ change? Will the stock on hand increase? How much?

  • (b) Assume that the lead-time is reduced to two weeks and that Q is unchanged. How should R be chosen to get the same S ₂ as before? How high is then the average stock on hand?

• 5.9 Consider a continuous review (R, Q) policy with R = 1 and Q = 2. The demand is Poisson with intensity λ = 0.5. The lead-time is 4.

  • (a) What is the average inventory level?

  • (b) What is the average inventory on hand?

  • (c) What is the average backorder level?

  • (d) What is the average waiting time for a customer?

• 5.10 5.10 Consider normal lead-time demand with mean 100 and standard deviation 30. An (R, Q) policy is applied with Q = 50 given. The holding cost is h = 2 per unit and unit time.

  • (a) Determine R so that the average waiting time for a customer is 0.01. Determine the corresponding holding costs.

  • (b) Choose S ₁, S ₂, and S ₃, respectively so that you get the same solution as in a).

  • (c) Choose a backorder cost b ₁ per unit and time unit so that you get the same solution as in a).

• 5.11* An inventory is controlled by an (R, Q) policy with continuous review. Demand that cannot be met directly is backordered. R = 1 and Q = 3. The demand per time unit has a Poisson distribution with mean 2. The lead-time is one time unit.

  • (a) What is the probability in steady state to have 2 units in stock?

  • (b) What is the average waiting time for a customer?

• 5.12* An item is controlled by an (R, Q) policy with continuous review. The lead-time demand has an exponential distribution with density:

  $$f(x) = \frac{1}{m}{e^{ - \frac{x}{m}}}\quad \;x \ge 0\quad $$

  where m is the mean. Determine the fill rate for given R, Q and m. Assume that R > 0.

• 5.13* Consider a continuous review (R, Q) policy with R = 1 and Q = 3. The demand is Poisson with intensity λ = 0.7. The lead-time is one time unit.

  • (a) Determine the fill rate S ₂ exactly.

  • (b) Determine the fill rate S ₂ by using a normal approximation.

• 5.14 Consider pure Poisson demand and a continuous review (s, S) policy. Use the technique in Sect. 5.11 to verify that the distribution of the inventory position is uniform.

• 5.15 Make a computer program to verify the results in Example 5.6.

• 5.16

  • (a) Consider the periodic review (R, Q) model in Section  5.12.2. Modify the derivation to fit an order-up-to-S policy.

  • (b) Use the results in (a) to determine the fill rate under the following assumptions. Poisson demand with mean 0.5 per unit of time, L = 1, T = 0.5, and S = 1.

• 5.17 Consider Sect. 5.12.4. Modify the derivation to fit an order-up-to-S policy.

• 5.18 Consider the newsboy model in Sect. 5.13 but assume a discrete demand distribution with distribution function F(x). Demonstrate that the optimality condition corresponding to (5.91) is to choose the smallest S satisfying F(Q) ≥ c _u /(c _o  + c _u ).

• 5.19 Use the result in Problem 5.18 to solve the following newsboy problem. Poisson demand with mean 3, c _o  = 2, and c _u  = 5. What are the optimal costs?

• 5.20 Consider a continuous review (R, Q) policy with R = 60 and Q = 80. The lead-time demand is normally distributed with mean 100 and standard deviation 20. Determine the fill rate both for complete backordering and for lost sales. Why is the fill rate higher in the lost sales case?

• 5.21* Consider Example 5.9 and the first case with a stochastic lead-time of type 1. Use (5.102) and (5.103) to find μ´ and σ´. Fit then a negative binomial distribution to these values. Using this distribution determine the probabilities for IL equal to 1, 2, 3, and compare to the exact solution.