Multi-Echelon Systems: Reorder Points

Abstract

This chapter deals with various techniques for determining safety stocks and reorder points in multi-echelon inventory systems. Throughout the chapter we assume that the batch quantities are given.

Problems

(*Answer and/or hint in Appendix 1)

• 10.1 Show that ${\hat{C}_2}({y_2})$ in (10.9) is convex in y ₂.

• 10.2* Consider Example 10.1. Assume that L ₂ = 0 instead of 5. Derive the corre- sponding expression for ${{\hat{C}}_{2}}({{y}_{2}})$. What is the optimal solution?

• 10.3 Consider Example 10.1. Set e ₁ = 0 and determine the corresponding optimal policy and costs.

• 10.4* Consider a two-level distribution system with one central warehouse, two re tailers, and Poisson demand at the retailers. All installations apply installation stock (S − 1, S) policies. We use the notation in Sect. 10.2. The transportation times are L ₀ = L ₁ = L ₂ = 1, and the demand intensities at the retailers are λ ₁ = 1 and λ ₂ = 2. The holding costs per unit and time unit are the same at all sites, h ₀ = h ₁ = h ₂ = 1. The backorder costs per unit and time unit are the same at both retailers b ₁ = b _2 = 10. The optimal policy is to use S ₀ = 3, S ₁ = 3, and S ₂ = 5. The exact holding costs per time unit are then 0.672 at the warehouse, 1.830 at retailer 1, and 2.627 at retailer 2. The backorder costs are 0.539 at retailer 1, and 0.752 at retailer 2. The total costs are 6.420 per time unit. The exact fill rates are 42.3 % at the warehouse, 87.0 % at retailer 1, and 88.8 % at retailer 2. Note the typical low warehouse fill rate.

  • a) Verify the exact holding costs and the exact fill rate at the warehouse.

  • b) Evaluate holding costs, backorder costs and fill rates at the retailers using the METRIC approximation.

• 10.5 Consider Problem 10.4. Introduce the constraint S ₀ = 0.

  • a) What does the constraint mean?

  • b) What is the optimal solution under this constraint?

• 10.6 We consider a two-level distribution system with one central warehouse and two retailers. The warehouse lead-time is 3 and the retailer lead-times are both equal to 1. The retailers face Poisson demand with intensities 0.1 (retailer 1) and 0.2 (retailer 2). All installations apply (S − 1, S) policies. The warehouse has S ₀ = 1, and the retailers have S ₁ = 0, and S ₂ = 1. Use the METRIC approximation to estimate the average number of backorders at the retailers in steady state.

• 10.7 Consider the serial inventory system in Fig. 10.1. The demand at installation 1 is Poisson with intensity 1. Both lead-times (transportation times) are equal to 2. The installations apply continuous review (S − 1, S) policies with S ₁ = 2, and S ₂ = 1. Use the METRIC approximation for estimating the average waiting time for a demand at installation 1.

• 10.8 Consider again the serial system in Fig. 10.1. Otherwise the assumptions are as in Sect. 10.2.

  • a) Reformulate the technique in Sect. 10.3.1 for this system.

  • b) Reformulate the technique in Sect. 10.3.2 for this system.

  • c) Set S ₂ = 0. Use the results in (a) and (b) for determining P(IL ₁ = 0). Verify that you get the same result. How can the solution be interpreted?

• 10.9 Make a computer program to determine the exact results in Problem 10.4 using the technique in Sect. 10.3.1.

• 10.10 Make a computer program to determine the exact results in Problem 10.4 using the technique in Sect. 10.3.2.

• 10.11 Consider the cost structure in Sect. 10.4. Explain why it can never be optimal to use S _i  < 0 for retailer i.

• 10.12 Consider the data in Example 5.8 in Sect. 5.13. Change the assumptions as in Sect. 10.6.2.3. Determine a buyback contract, which will maximize the total profit and give each firm 50 % of the total profit.