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Introduction Background Information Problem Description ILP Formulation Results Conclusion
Press Sheet Optimization for Open Loop Controlof Industrial Scale Gang-Run Printing
Daniel FullmerSean Warnick
IDeA LabsBrigham Young University
March 10, 2014
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Introduction
Background Information
Problem Description
ILP Formulation
Results
Conclusion
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Introduction
I Factory optimization problem faced by the printing industry.
I For each day, the problem is to choose which orders to printand the press sheets to print them on that would minimize thetotal cost of production.
I We formulate this problem as an integer linear programmingproblem and use a common solver.
I Our method was compared with optimization by hand in areal factory and significant cost savings were found.
I High order volume means that small improvements inefficiency can lead to large cost savings.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Printed Products
t ∈ T , T = {‘business cards’, ‘4x6 postcards’, . . . }
Orders may be 1 or 2-sided.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Press Sheets
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Press Sheet Templates
p ∈ P, P = {‘18x46PC’, ‘63xBC’, ‘9x46PC-36xBC’, . . . }
The capacity of a particular template for a product type is givenby: upt
u‘18x46PC’,‘4x6 postcards’ = 18
u‘18x46PC’,‘business cards’ = 0
u‘63xBC’,‘4x6 postcards’ = 0
u‘63xBC’,‘business cards’ = 63
u‘9x46PC-36xBC’,‘4x6 postcards’ = 9
u‘9x46PC-36xBC’,‘business cards’ = 36
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
“Traditional” Printing
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Gang-Run Printing
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Cutting
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Presses
m ∈ M, M = {‘Offset Press 1’, ‘Digital Press 1’, . . . }
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Digital Presses
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Offset Presses
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Offset Press Technology
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
UV Coating
Three types of UV Coating:
I UV Spot
I UV Full (Flood)
I UV None
Spot coating can be used for orders needing Spot, Full, or None.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
UV Spot Coating
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Attributes
a ∈ A, A = {2SS, 2SF, 2SN, 2FF, 2FN, 2NN, 1S, 1F, 1N}
These attributes support any combination of UV front/back byflipping.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Problem Description
I Each day, choose:I The orders to printI The press sheets to print them onI How to place those orders onto those press sheets
I Industry Intuition: Wait as long as possible to print an orderso it has the best chance of being placed efficiently.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Orders
I Typically fixed at certain quantities:
q ∈ Q, Q = {. . . , 500, 1000, 2000, 2500, . . . }
I Order types are defined by a 3-tuple inclding a product type t,attributes a, and quantity q: (t, a, q).
I Required orders: vtaqI Optional orders: wtaq
I Example: v‘business cards’,1N,1000 = 121
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Press Sheets Revisited
I For our purposes: We define a press sheet by a 4-tupleincluding a template p, attributes a, press m, and quantity q:(p, a,m, q).
I Number of press sheet template p to produce with attributesa on machine m at quantity q: bpamq
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Order Placement Complexity
I Multiple ups: If an order is placed n times on a press sheet,we say it is n-up on that press sheet.
I Overprinting: The a customer ordered 750 business cards, wecan print 1000 and discard 250.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Cost Summary
Digital Press Offset Press
Press Size Smaller LargerPress speed Slow Fast
Ink Expensive CheapPlates None Typically 4
Setup time None SignificantEfficient Quantities Low High
I Ultimately, Digital presses have better per-run costs, butworse per-sheet costs.
I UV Spot coating is relatively expensive.
I Cutting Labor: prefer simple press sheet templates
The cost of producing a press sheet is given as: cpamq
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Best Possible Cost for an Order
I We prefer to place every order 1-up on a press sheet that hasthe same attributes and quantity as the order being placed.
I Best possible cost: (where zt is the area of product type t)
ktaq = minm∈M, p∈Mp
cpamqzt∑t′∈T upt′zt′
I This provides a lower bound on the cost of producing aparticular product.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Integer Linear Programming Formulation
The solution is split into two parts:
I Solving an Integer Linear Program (ILP)
I A simple postprocessing step that “executes” the ILP solution.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Objective Function
Optimize the cost of the press sheets we build, along with a lowerbound on the cost of unprinted orders.
minb,d ,d ′,d ′′
r ,r ′,r ′′,r ′′′
∑p∈P, a∈A
m∈Mp , q∈Q
cpamqbpamq +∑t∈Ta∈Aq∈Q
ktaq(vtaq + wtaq − rtaq)
cpamq Cost of producing press sheetbpamq Number of press sheet jobs to producektaq Best possible costvtaq Number of required orderswtaq Number of optional ordersrtaq Number of orders to print
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Order selection
subject to
vtaq ≤rtaq ≤ vtaq + wtaq
vtaq Number of required orderswtaq Number of optional ordersrtaq Number of orders to print
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Overprinting
rtaq =∑qto∈Qq≤qto
dtaqqto
r ′taq =∑
qfrom∈Qqfrom≤q
dtaqfromq
rtaq Number of orders to printr ′taq Number of orders to print (after overprinting)
dtaqfromqto Number of times to treat orders with quantity qfromas an order with quantity qto
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Quantity Splitting
r ′′taq = r ′taq −∑qto∈Q
(q−qto)∈Qq/2≤qto<q
d ′taqqto +∑
qfrom∈Q(q−qfrom)∈Q
qfrom/2≤q≤qfrom
(d ′taqfromq + d ′taqfrom(q−qfrom)
)
r ′taq Number of orders to print (after overprinting)r ′′taq Number of orders of product to print (after splitting)
d ′taqfromqto Number of times to treat orders with quantity qfromas an order with quantity qto
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Supporting Attributes
Allow orders to be printed on press sheets that support them:
r ′′taq =∑
a∈Aato
d ′′taatoq
r ′′′taq =∑
afrom∈Aa
d ′′tafromaq
r ′′taq Number of orders of product to print (after splitting)r ′′′taq Number of slots to print (after attributes)
d ′′taqfromqto Number of times to treat orders with quantity qfromas an order with quantity qto
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Press Sheet Assignment
Make sure we have enough positions on press sheets to support theorders to be printed:
r ′′′taq ≤∑p∈Pm∈Mp
uptbpamq
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Postprocessing
I Simply follow the instructions from a feasible ILP solution.(bs, ds, and rs).
I Create enough press sheets according to b.
I Use the ds to determine how orders are distributed out topress sheets.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Simulation
I Using real data from a factory
I Comparison with existing manual order assignment
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Simulation Results
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Day
Sca
led
Cos
t
Manual OptimizedDaniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Possible Future Work
1. Don’t constrain the press sheets to be printed at fixedquantities.
2. Include 2D rectangle packing with guillotine cuts.
3. Do smarter things with postprocessing. Color balancing,placing similar orders together, etc.
4. Try to smoothing production volume over multiple days withpredictions about future orders.
5. Consider the possibility of failed production.
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
Introduction Background Information Problem Description ILP Formulation Results Conclusion
Thank you!
Daniel Fullmer IDeA Labs Brigham Young University
Press Sheet Optimization for Open Loop Control of Industrial Scale Gang-Run Printing
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