press sheet optimization for open loop control of ...€¦ · ‘18x46pc’,‘business cards’ =...

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

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.




Printed Products

t ∈ T , T = {‘business cards’, ‘4x6 postcards’, . . . }

Orders may be 1 or 2-sided.




Press Sheets




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




“Traditional” Printing




Gang-Run Printing




Cutting




Presses

m ∈ M, M = {‘Offset Press 1’, ‘Digital Press 1’, . . . }




Digital Presses




Offset Presses




Offset Press Technology




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.




UV Spot Coating




Attributes

a ∈ A, A = {2SS, 2SF, 2SN, 2FF, 2FN, 2NN, 1S, 1F, 1N}

These attributes support any combination of UV front/back byflipping.




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.




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




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




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.




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




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.




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.




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




Order selection

subject to

vtaq ≤rtaq ≤ vtaq + wtaq

vtaq Number of required orderswtaq Number of optional ordersrtaq Number of orders to print




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




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




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




Press Sheet Assignment

Make sure we have enough positions on press sheets to support theorders to be printed:

r ′′′taq ≤∑p∈Pm∈Mp

uptbpamq




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.




Simulation

I Using real data from a factory

I Comparison with existing manual order assignment




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



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.




Thank you!



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