Recoverable Service Parts Inventory Problems

-Ibrahim Mohammed

IE 2079

Overview

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

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.

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?

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?

Applications

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

Motivation

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

Echelon Re-supply Structure

Depot based Re-supply Systemλij Rate of removal of LRU i at base j (Poisson process)

rij Probability of LRU i being replaced at base j

Bij Base repair cycle time for LRU i at base j

Aij Order, shipping and receiving time of LRU i at base j

Di Depot repair cycle time for LRU i

Sij Stock level for LRU i at base j

Sio Stock level for LRU i at depot.

Mathematical Formulation

Βij Backorder of LRU i at base j

b Total budget available ci Cost of one unit of LRU i

m Total no of bases

i

ioijij

m

j

ssMin )/(1

bsscSTi

ij

m

jioi

)(.1

This formulation cannot be solved using simplex as the objective is non linear.

Minimize backorders at base. (Backorders is a function of sij and sio. It can be solved using queuing models or simulation).

Total inventory purchased should be within the allotted budget

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.

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.

Order upto level at base, Sj = Inventory at base repair + Inventory at base stock + Inventory in transit from the depot to the base

Order upto level at depot, SD = Inventory at depot repair + Inventory at depot stock + Inventory shipped from base to depot

Optimum policy : Determine all Ss, such that

S1+ S2+ S3+….+Sm + SD = C (total inventory constant)

Optimum Policy

Average LRU i resupply time at base j:

Expected depot delay

Expected no of base j backorders for LRU i

where, Xij is the no of units in re-supply for LRU i at base j in

steady state.

))()(1( ioijijijijij sArBrT

io

ioio

ss

)(

)(

?ijXE

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

Example Problem

Given the following, what is the expected no of units in re-supply for the base?

Rij = o, Aij = 5, λij = 5, λio = 50, δ(sio) = 0.0563 & Di = 1

Solution:

= 5[0 + 1(5+0.0563)] = 25.2815

ijijij TXE

))]()(1([][ ioijijijijijij sArBrXE