1. Recoverable Service Parts Inventory Problems -Ibrahim Mohammed IE 2079

2. Overview Definitions Types of Decisions Applications Motivation Taxonomy of Service Parts Inventory System Problem Definition Mathematical Formulation METRIC Little’s Law Example Problem

3. Definitions Inventory system – A set of policies and controls that monitors levels of inventory and determines what levels should be maintained, and how orders should be met. Service Parts – Used as a replacement for defective components or parts. Recoverable item – Item which can be fixed or repaired. LRU – Line replaceable unit.

4. Types of Decision Strategic What are the customer requirements & how to allocate resources to meet these requirements? Operational What do I repair now and what do I ship now from one location to another via what mode of transport?

5. Types of Decisons Tactical What inventories will be needed to meet operational objectives at some future time given the design and operational characteristics of an existent re-supply system infrastructure?

6. Applications Applications: Automotive Industry Space Stations Airline Industry Power Plants (Eg: Nuclear, Coal plants etc..)

7. Motivation

8. Taxonomy of Service Parts Inventory Systems Echelon or Network Re-supply Structure Service parts are consumable Demand is high Cheap Depot Base Re-supply System Service parts are repairable Demand is low Very expensive

9. Echelon Re-supply Structure

10. Depot based Re-supply System λ ij  Rate of removal of LRU i at base j (Poisson process) r ij  Probability of LRU i being replaced at base j B ij  Base repair cycle time for LRU i at base j A ij  Order, shipping and receiving time of LRU i at base j D i  Depot repair cycle time for LRU i S ij  Stock level for LRU i at base j S io  Stock level for LRU i at depot.

11. Mathematical Formulation Β ij  Backorder of LRU i at base j b  Total budget available c i  Cost of one unit of LRU i m  Total no of bases This formulation cannot be solved using simplex as the objective is non linear. Minimize backorders at base. (Backorders is a function of s ij and s io. It can be solved using queuing models or simulation). Total inventory purchased should be within the allotted budget

12. Problem Definition Purchase minimum amount of inventory within the allotted budget, and stock the inventory at the depot and the various bases j, such that the expected no of backorders at the bases is minimized at any given point in time.  How much should we invest in inventory?  Where should service parts be stocked and in what quantities? Proposed Solution:  The optimum policy would be to store all the inventory at the depot. This is so because the variance of demand distribution decreases when all the goods are stored at a central location. Therefore, the overall inventory required to meet the demand would be greatly reduced.  The problem with this approach is however that the lead time to service a component would increase as the service part would have to be shipped out from the depot to the base all the time.

13. METRIC Definition: METRIC is a mathematical model translated into a computer program, capable of determining base and depot stock levels for a group of recoverable items; its governing purpose is to optimize system performance for specified levels of system investment. Purposes: Optimization: Determine the optimal base and depot stock levels for each line item. Redistribution: Optimally allocate the total stock between the bases and the depot. Evaluation : Provides an assessment of the performance and investment cost for the system of any allocation of stock between the bases and depot. Assumption All parts are repairable. Therefore, there is never a net loss in inventory of the system. Depot meets demand on a first-come first-serve basis. Bases are not re-supplied by other basis. Repair cycle time is not dependent on base from which the LRU is sent.

14. Order upto level at base, S j = Inventory at base repair + Inventory at base stock + Inventory in transit from the depot to the base Order upto level at depot, S D = Inventory at depot repair + Inventory at depot stock + Inventory shipped from base to depot Optimum policy : Determine all S s, such that S 1 + S 2 + S 3 +….+S m + S D = C (total inventory constant)

15. Optimum Policy Average LRU i resupply time at base j: Expected depot delay Expected no of base j backorders for LRU i where, X ij is the no of units in re-supply for LRU i at base j in steady state.

16. Little’s Law The average number of customers in a queuing system N is equal to the average arrival rate of customers to that system λ, times the average time spent in that system, t. Therefore, N = λ x t

17. Example Problem Given the following, what is the expected no of units in re-supply for the base? R ij = o, A ij = 5, λ ij = 5, λ io = 50, δ(s io ) = 0.0563 & D i = 1 Solution: = 5[0 + 1(5+0.0563)] = 25.2815