Flexible Assembly Systems

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Topic 28

• Flexible Assembly Systems

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Job Shops Flexible Assembly

• Each job has an unique identity
• Make to order, low volume environment
• Possibly complicated route through system
• Very difficult

• Limited number of product types
• Given quantity of each type
• Mass production
• High degree of automation
• Even more difficult!

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Flexible Assembly Systems
Sequencing Unpaced Assembly Systems Simple flow
line with finite buffers Application assembly of
copiers Sequencing Paced Assembly
Systems Conveyor belt moves at a fixed
speed Application automobile assembly Scheduling
Flexible Flow Systems Flow lines with finite
buffers and bypass Application producing printed
circuit board
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Sequencing Unpaced Assembly Systems

• Number of machines in series
• No buffers
• Material handling system
• When a job finishes moves to next station
• No bypassing
• Blocking
• Can model any finite buffer situation

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Cyclic Schedules

• Schedules often cyclic or periodic
• Given set of jobs scheduled in certain order
• Contains all product types
• May contain multiple jobs of same type
• Second identical set scheduled, etc.
• Practical if insignificant setup time
• Low inventory cost
• Easy to implement

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Minimum Part Set

• Suppose l product types
• Let Nk be target number of jobs of type k
• Let z be the greatest common divisor
• Then
• is the smallest set with correct proportions
• Called the minimum part set (MPS)

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Defining a Cyclic Schedule

• Consider the jobs in the MPS as n jobs
• Let pij be as before
• A cyclic schedule is determined by sequencing the
  job in the MPS
• Maximizing TP Minimizing cycle time

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MPS Cycle Time Example
Buffer!
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Sequence 1,2,3
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Cycle Time 3
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Sequence 1,3,2
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Cycle Time
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Minimizing Cycle Time

• Profile Fitting (PF) heuristic
• Select first job j1
• Arbitrarily
• Largest amount of processing
• Generates profile
• Determine which job goes next

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PF Next Job

• Compute for each candidate job
• Time machines are idle
• Time job is blocked
• Start with departure times

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Nonproductive Time

• Calculate sum of idle and blocked time
• Repeat for all remaining jobs in the MPS
• Select job with smallest number
• Calculate new profile and repeat

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Discussion PF Heuristic

• PF heuristic performs well in practice
• Refinement
• Nonproductive time is not equally bad on all
  machines
• Bottleneck machine
• Use weight in the sum

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Discussion PF Heuristic

• Basic assumptions
• Setup is not important
• Low WIP is important
• ? Cyclic schedules good
• Want to maximize throughput
• ? Minimize cycle time
• ? PF heuristic performs well

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Discussion FMS

• Flexible Manufacturing Systems (FMS)
• Numerically Controlled machines
• Automated Material Handling system
• Produces a variety of product/part types
• Scheduling
• Routing of jobs
• Sequencing on machines
• Setup of tools
• Similar features but more complicated

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Discussion Solution Methods

• Formulated as simple sequencing
• Can apply branch-and-bound
• In general constraints make mathematical
  programming formulation difficult
• PF heuristic easy to generalize

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Additional Complications

• The material handling system does not wait for a
  job to complete
• ? Paced assembly systems
• There may be multiple machines at each station
  and/or there may be bypass
• ? Flexible flow systems with bypass

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Topic 29

• Paced Assembly Systems

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Paced Assembly Systems

• Conveyor moves jobs at fixed speeds
• Fixed distance between jobs
• Spacing proportional to processing time
• No bypass
• Unit cycle time
• time between two successive jobs

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Grouping and Spacing

• Attributes and characteristics of each job
• color, options, destination of cars
• Changeover cost
• Group operations with high changeover
• Certain long operations
• Space evenly over the sequence
• Capacity constrained operations (criticality
  index)

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Objectives

• Minimize total setup cost
• Meet due dates for make-to-order jobs
• Total weighted tardiness
• Spacing of capacity constrained operations
• Pi(l) penalty for working on two jobs l
  positions apart in ith workstation
• Regular rate of material consumption

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Grouping and Spacing Heuristic

• Determine the total number of jobs to be
  scheduled
• Group jobs with high setup cost operations
• Order each subgroup accounting for shipping dates
• Space jobs within subgroups accounting for
  capacity constrained operations

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Example

• Single machine with 10 jobs
• Each job has a unit processing time
• Setup cost
• If there is a penalty cost

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Example Data
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Grouping

• Group A Jobs 1,2, and 3
• Group B Jobs 4,5, and 6
• Group C Jobs 7,8,9, and 19
• Best order A ? B ? C

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Grouped Jobs
A B C
Due date
? Order A ? C ? B
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Capacity Constrained Operations
A C B
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Topic 30

• Flexible Flow Systems

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Flexible Flow System with Bypass
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Flexible Flow Line Algorithm

• Objectives
• Maximize throughput
• Minimize work-in-process (WIP)
• Minimizes the makespan of a days mix
• Actually minimization of cycle time for MPS
• Reduces blocking probabilities

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Flexible Flow Line Algorithm

• Three phases
• Machine allocation phase
• assigns each job to a specific machine at station
• Sequencing phase
• orders in which jobs are released
• dynamic balancing heuristic
• Time release phase
• minimize MPS cycle time on bottlenecks

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Machine Allocation

• Bank of machines
• Which machine for which job?
• Basic idea workload balancing
• Use LPT dispatching rule

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Sequencing

• Basic idea spread out jobs sent to the same
  machine
• Dynamic balancing heuristic
• For a given station, let pij be processing time
  of job j on ith machine
• Let

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Dynamic Balancing Heuristic

• Let Sj be the jobs released before and including
  job j
• Define
• Target

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Minimizing Overload

• Define the overload of the ith machine
• The cumulative overload is
• Minimize

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Release Timing

• MPS workload of each machine known
• Highest workload bottleneck
• MPS cycle time ? Bottleneck cycle time
• Algorithm
• Step 1 Release all jobs as soon as possible
• Step 2 Delay all jobs upstream from bottleneck
  as much as possible
• Step 3 Move up all jobs downstream from the
  bottleneck as much as possible

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Example
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Data
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Machine Allocation
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Workload
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Overload
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Overload Matrix
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Dynamic Balancing
First Job
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Selecting the Second Job

• Calculate the cumulative overload
• where

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Cumulative Overload
Selected next
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Final Cycle

• Schedule jobs 4,5,1,3,2
• Release timing phase
• Machine 4 is the bottleneck
• Delay jobs on Machine 1, 2, and 3
• Expedite jobs on Machine 5

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Topic 31

• Lot Sizing

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Lot Sizing

• Domain
• large number of identical jobs
• setup time/cost significant
• setup may be sequence dependent
• Terminology
• jobs items
• sequence of identical jobs run

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Applications

• Continuous manufacturing
• chemical, paper, pharmaceutical, etc.
• Service industry
• retail procurement

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Objective

• Minimize total cost
• setup cost
• inventory holding cost
• Trade-off
• Cyclic schedules

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Scheduling Decisions

• Determine the length of runs
• lot sizes
• Determine the order of the runs
• sequence to minimize setup cost
• Economic Lot Scheduling Problem (ELSP)

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Overview

• One type of item/one machine
• Several types of items/one machine
• rotation schedules
• arbitrary schedules
• Generalizations to multiple machines

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Problem Description

• Single machine
• Single item type
• Production rate q/time
• Demand rate g/time
• Problem determine the run length

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Minimize Cost

• Let x denote the cycle time
• Demand over a cycle gx
• Length of production run needed gx/q

Inventory
Time
x
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Costs
Setup cost per run

• Average setup cost c/x
• Average inventory holding cost
• Total cost

Per item holding cost
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Optimizing Cost

• Derivative
• Solve

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Optimal Cycle Time
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Optimal Lot Size

• Total production
• When unlimited production capabilities
• Economic Order Quantity (EOQ)

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Setup Time

• Setup time s
• If s ? x(1-r) above optimal
• Otherwise cycle length
• is optimal

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Numerical Example

• Production q 90/month
• Demand g 50/month
• Setup cost c 2000
• Holding cost h 20/item

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Optimal Schedule

• Cycle time 3 months
• Lot size 150 items
• Idle time 3(1-5/9)1.33 months

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Example Setup Times

• Now assume setup time
• If lt 1.33 months then 3 month cycle still optimal
• Otherwise the cycle time must be longer

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Inventory Levels
Inventory
Month
1 2 3 4 5
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Inventory
Month
1 2 3 4 5
6
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Example

• A plant needs to produce 10000 car chassis per
  year
• The plant capacity is 25000 chassis/year
• Each chassis costs 2000
• It costs 200 to set up a production run
• Holding cost is 500/chassis/year
• What is the optimal lot size?

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Solution

• The optimal lot size is
• which means we should make
• runs in a year.

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Discussion

• Notice that the preceding result does not tell us
  how to produce those chassis in detail
• Lot size models are used for planning
• Time horizon usually a few months (short range
  planning)

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Topic 33

• Lot Sizing with Multiple Items

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Multiple Items

• Only considered one item type before
• Now assume n different items
• Demand rate for item j is gj
• Production rate of item j is qj
• Setup independent of the sequence
• Rotation schedule single run of each item

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Scheduling Decision

• Cycle length determines the run length for each
  item
• Only need to determine the cycle length x
• Expression for total cost/time unit

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Inventory Holding Cost

• Average inventory level for the j-th item
• Average total cost

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Optimal Lot-Size

• Solve as before
• Limiting case (infinite production rate)

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Example
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Solution
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Solution

• The total average cost per time unit is
• How can we do better than this?

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Topic 34

• Lot Sizing with Setup

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Setup Times

• With sequence independent setup costs and no
  setup times the sequence within each lot does not
  matter
• ? Only a lot sizing problem
• Even with setup times, if they are not job
  dependent then still only lot sizing

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Job Independent Setup Times

• If sum of setup times lt idle time then our
  optimal cycle length remains optimal
• Otherwise we take it as small as possible

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Job Dependent Setup Times

• Now there is a sequencing problem
• Objective minimize sum of setup times
• Equivalent to the Traveling Salesman Problem
  (TSP)
• A salesman must visit n cities exactly once with
  the objective of minimizing the total travel
  time, starting and ending in the same city

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Equivalence to TSP

• Item city
• Travel time setup time
• TSP is NP-hard
• If best sequence has sum of setup times lt idle
  time ? optimal lot size and sequence

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Long setup

• If sum of setups gt idle time, then the optimal
  schedule has the property
• Each machine is either producing or being setup
  for production
• An extremely difficult problem with arbitrary
  setup times

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Arbitrary Schedules

• Sometimes a rotation schedule does not make sense
  
• (remember problem with no setup cost)
• For example, we might want to allow a cycle
  1,4,2,4,3,4 if item 4 has no setup cost
• No efficient algorithm exists

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Problem Formulation

• Assume sequence-independent setup
• Formulate as a nonlinear program

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Notation

• Setup cost and setup times
• All possible sequences
• Item k produces in l-th position
• Setup time sl, run time tl, and idle time ul

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Inventory Cost

• Let x be the cycle time
• Let v be the time between production of k
• Total inventory cost for k is

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Mathematical Program
Subject to
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Two Problems

• Master problem
• finds the best sequence
• Subproblem
• finds the best production times, idle times, and
  cycle length
• Key idea think of them seperately

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Subproblem
Subject to
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Master Problem

• Sequencing complicated
• Heuristic approach
• Frequency Fixing and Sequencing (FFS)
• Focus on how often to produce each item
• Computing relative frequencies
• Adjusting relative frequencies
• Sequencing

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Computing Relative Frequencies

• Let yk denote the number of times item k is
  produced in a cycle
• We will
• simplify the objective function by substituting
• drop the second constraint

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Mathematical Program
Subject to
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Solution

• Using Lagrangean multiplier
• Adjust cycle length for frequencies
• Idle times l 0
• No idle times, must satisfy

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Adjusting the Frequencies

• Adjust the frequencies such that they are
• integer
• powers of 2
• cost within 6 of optimal cost
• New frequencies and run times

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Sequencing

• Variation of LPT
• Calculate
• Consider the problem with machines in
  parallel and jobs of length
• List pairs in decreasing order
• Schedule one at a time considering spacing

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Topic 35

• Lot Sizing on Multiple Machines

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Multiple Machines

• So far, all models single machine models
• Extensions to multiple machines
• parallel machines
• flow shop
• flexible flow shop

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Parallel Machines

• Have m identical machines in parallel
• Setup cost only
• Item process on only one machine
• Assume
• rotation schedule
• equal cycle for all machines

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Decision Variables

• Same as previous multi-item problem
• Addition assignment of items to machines
• Objective balance the load
• Heuristic LPT with

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Different Cycle Lengths

• Allow different cycle lengths for machines
• Intuition should be able to reduce cost
• Objective assign items to machines to balance
  the load
• Complication should not assign items that favor
  short cycle to the same machine as items that
  favor long cycle

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Heuristic Balancing

• Compute cycle length for each item
• Rank in decreasing order
• Allocation jobs sequentially to the machines
  until capacity of each machine is reached
• Adjust balance

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Further Generalizations

• Sequence dependent setup
• Must consider
• preferred cycle time
• machine balance
• setup times
• Unsolved
• General schedules ? even harder!
• Research needed -)

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Flow Shop

• Machines configured in series
• Assume no setup time
• Assume production rate of each item is identical
  for every machine
• ? Can be synchronized
• Reduces to single machine problem

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Variable Production Rates

• Production rate for each item not equal for every
  machine
• Difficult problem
• Little research
• Flexible flow shop need even more stringent
  conditions

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Discussion

• Applicability of lot sizing models
• short range planning
• demand assumed known
• determines throughput
• make-to-stock systems
• due date of little importance/not available
• extensions to mixed systems
• Multiple facilities in series
• supply chain management