Instantaneous Service Level: probability that a spare is available at any moment
Service Level in an Interval (or mission): probability of not running out of stock at any time over a specific time interval
Global Cost: it is the most used criterion. Includes: Acquisition Costs, Intervention Costs, Spare Parts, Ownership Costs, Failure Costs
Industry Context
Hard Times
PCR
Planf. & Prog.
Replacement Components Main
Features Impact Minimize Unscheduled Stops
Frequency according to MTBF & MTBS
Guidelines with measurement and control tasks
Construction Preventive Plan
Who is in control?
Should be minimized
Negative influence on MTBF, MTBS, MTTR & Availability
Review of Strategies, Plans and Frequencies according to the Failure Modes (RCM)
MTBF, MTBS, MTTR, Availability & Maintenance Plan Fixed Frequency
Well-known routines
Management Base
Efficient / Effective Maintenance
Availability
A window to plan equipment maintenance to a certain standard
Frequency determined by the Engineering Area
Maintenance Ability to Maintain the Maintenance Plan
Frequency Determined by Life Objectives
Use window to restore equipment to standard
Starting point of a new Life Cycle
Component
Costs (US $)
Equipment Maintenance & Repair
Inspection routines
Non-Scheduled Repairs
Periodic Services (PM)
Major & Minor Repairs Program
Reasons Stopped
The way to follow……
PHASE 0: Understanding PROBLEM (S) - Types of Inventories
PHASE 1: Re-Design of STRATEGY
• Mission - Vision
• Strategic Objectives
PHASE 2:
PHASE 3:
PHASE 4:
PHASE 5:
PHASE 6: PHASE 7: • Process Design Consumable - Repairable Inventories
• Procedures
• Organization chart - ROLE & RESPONSIBILITIES • Metrics - KPI's
• Others…..
Team Review - “Technical Competences” - TRAINING DECISION CRITERIA - Policies
DEVELOPMENT OF OPTIMIZATION MODELS (Single - echelon / Multi -echelon) SIMULATION MODELS
SAP CONFIGURATION / Others
NEGOTIATIONS - Agreements with Suppliers / Factory
Repairable Inventory Management Problem Repairable
Components
The Problem
Demand for Failures - Company
Lawsuit Components “XX”
Performance measures
• The Cycle Service Level (CSL) that indicates the percentage of cycles in which there are no stock breaks, also known as ?? 1.
• The Fill Rate (FR) which is defined as the fraction of demand that is covered by the available physical stock or ?? 2.
• The Ready Rate or fraction of time during which the net stock is positive, also known as ?? 3.
• Backorders (BO) number of demands that are not satisfied at any point of time.
• The Average Time Between Stock Breakouts (TBS).
Optimization Criteria
• Instant Service Level: probability that a spare is available at any moment.
• Service Level in an Interval (or mission): probability of not running out of stock at any time over a specific time interval.
• Global Cost: it is the most used criterion.
Includes:
o Acquisition Costs o Spare Parts Intervention Costs o Property Costs o Failure Costs
• Availability of the Supported System: fraction of the time the equipment is in service as a result of the availability of spare parts.
• Operational Availability: we assume that each backordered component results in a non-operational system.
Deterministic Models (EOQ type)
Inventory System with Repair
Models Repairable Inventories
Case 1: Case 2:
Models Repairable Inventories
Case 3:
Other cases ………
Stochastic Models
Cases under Study ……
System Assumption Criteria Optimization Cases / Policy
Single - Echelon Problem Capacity Unlimited Repair
Instant Reliability 1.- One-to-one repair (one-for-one)
2.- Batch repair
Whole Lot Regardless of Size
Whole Lot Regardless of Size - Identical Repair Rate
Repair by Lot of Specific
Size Size of Initial Inventory Greater than Repair Lot
Size of Initial Inventory Less than Repair Lot
Minimization Expected Backorders Subject to a Inventory Investment Restriction
Availability Maximization 1.- Operational
2.- Supported
Costs 1.- Downtime costs, inventory maintenance
costs 2.- Total
costs 3.- Total costs subject to service restrictions
Maximization Expected Fill Rate Subject to Inventory Investment Restriction
Limited Repair Capacity Instant Reliability 1.- Number of repair channels ≤ number of fleet units
2.- Number of repair channels> number of fleet units
Thank you ……… Back-up
Component Repair Process
Case: Instant Reliability, One-for-one Repair
Case: Maximizing Supported Availability
Case: Minimization of Expected Backorders Subject to Inventory Investment Restriction
Inventory Position
???? = ?????? - ???? + ?????? + ???? - ??????
???? = inventory position at time t
?????? = number of available units (on-hand) in time t
???? = number of units that fail in time t
?????? = number of units in the repair queue system at time t
???? = number of units in the supplier's order system at time t
?????? = number of units decommissioned at time t
?????? = ?????? - ???? <0 number of pending units (Backorders) in time t ???? = net inventory at time t
???? = ???? + ?????? + ??????
Louit, D., Pascual, R., Banjevic, D., Jardine, AKS,
Optimization Models for Critical Spare Parts Inventories - A Reliability Approach. Working paper, University of Toronto, 2005.
Sherbrooke, CC Optimal Inventory Modeling of Systems: multi-echelon techniques, Second Edition, Kluwer, Bostos, 2004.
Muckstadt, J. and Sapra, A., Principles of Inventory Management: When you are Down to four, order more. Springer, 2010.
The problem
Let ?? (??) be a random variable representing the number of units under repair (replenishment) at some arbitrary time ??. We will distinguish between the “backorder” case in which «??», has the range 0 ≤ ?? <∞, and the “lost sales” case where «??» is restricted to range 0 ≤ ?? <??. In the “lost sales” case, any demand that occurs when they exist? units in resupply is rejected, since there is no stock on hand.
We will use a continually revised “Inventory Policy (?? - 1, ??)”. Be "??" the demand rate of the customer order process.
Backorder case
Theorem: Let «S» be the stock level for an item whose demand is generated by a Poisson process with rate «λ». Consider that the replenishment time is a random variable with density function g (t) with mean «T» and distribution function G (t). Suppose resupply times are independent and identically distributed across customer orders. Then the steady-state probability that "x" units are replenished is given by
(λT) x
h (x) = e − λT x!
Lost Sales Case
Theorem: Suppose that customer orders arrive according to a Poisson Process with arrival rate λ. In addition, suppose the Stock level is S, and the replenishment time for accepted customer orders are independently and identically distributed with common density gτ = βe-βτ, with mean τ = 1β. Then the stationary probability that x units are under repair in the case of lost orders is given by
x
e − λββλ / x! e − λτ (λτ) x / x! πx = n = S () n
∑S e βλ ∑n = 0e − λτλτn! −λβ
n!
n = 0
The problem
Let ???? (??) be the probability that «??», (?? = 0.1,……., ??), machines are under repair at time «??». Let ?? (??) = the state probability vector at time t. The probability vector ?? (??) satisfies the system of differential equations
• ?? (??): number of units, “??”, subjected to repair at a particular moment in time “??”.
Yes ?? is the initial size of the inventory stock, we obtain the following feasible cases:
• Yes ?? ≤ ??, the number of units in operation remains “??”, and the current stock size is “?? - ?? ”.
Yes ?? > ??, the number of units in operation “?? + ?? - ?? ”, and the stock is out of stock.
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