Carnegie Mellon University - Planning and …egon.cheme.cmu.edu/ewo/docs/PPGrlima_ewo_march_2010.pdfCarnegie Mellon University EWO meeting, March 2010 - p. 2 Project description Objectives:

Planning and scheduling of PPG glass production,model and implementation.

Ricardo LimaIgnacio [email protected]

Carnegie Mellon University

Yu JiaoPPG Industries

Glass Business and Discovery Center

Carnegie Mellon University EWO meeting, March 2010 - p. 2

Project description

Objectives:

1. Development of a Mixed Integer Linear Programming (MILP) model for theplanning and scheduling of the glass production

◆ capture the essence of the process that is not considered in the Master Production Schedule(MPS)

◆ management of waste glass (cullet)

2. Current focus: implement the model into a user-friendly app lication based on Excel andGAMS


Project description

Objectives:

1. Development of a Mixed Integer Linear Programming (MILP) model for theplanning and scheduling of the glass production

◆ capture the essence of the process that is not considered in the Master Production Schedule(MPS)

◆ management of waste glass (cullet)

2. Current focus: implement the model into a user-friendly app lication based on Excel andGAMS

Research challenges

1. Integration of planning and scheduling into one model for a real process

2. Develop a model that provides solutions in reasonable computational time

3. Integrate the scheduling model with the waste glass management


Process and products

Continuous process:

Products defined by color

Substrate Coated

Solexia

Caribia

Azuria

Solarbronze

Solargray

Graylite



Continuous process:


Substrate Coated

Solexia

Caribia

Azuria

Solarbronze

Solargray

Graylite

Features◆ Raw materials define color of the substrate

◆ Sequence dependent changeovers betweensubstrates

◆ No transition times between substrate andcoated products



Continuous process:


Substrate Coated

Solexia

Caribia

Azuria

Solarbronze

Solargray

Graylite

Features◆ Raw materials define color of the substrate

◆ Sequence dependent changeovers betweensubstrates

◆ No transition times between substrate andcoated products

◆ Long transition times (order of days)

◆ High transition costs

◆ Continuous operation during changeover

◆ Minimum run length (days)


Process and cullet management

Cullet: waste glass

Good cullet: generated during the production run

Dilution cullet: produced during changeover from one substrate to other substrate


Problem statement

Given:◆ Time horizon of 18 months◆ Transition, and inventory costs◆ Set of products

◆ deterministic demand◆ initial, minimum, and maximum inventory levels◆ production rates◆ sequence dependent transitions◆ operating costs◆ selling prices

◆ Cullet◆ initial, minimum, and maximum inventory levels◆ production and consumption rates◆ compatibility matrix between colors◆ selling price

Determine:◆ sequence of production (production times and amounts)◆ inventory levels of products during and at the end of the time horizon◆ inventory levels of cullet during and at the end of the time horizon◆ economic terms: total operating, transition, inventory costs

That maximize the profit


Why is important to consider cullet?

Cullet management defines the feasible space of operation

◆ The amount and quality of cullet produced and consumed is related with the schedule.

◆ The cullet in stock must meet the schedule requirements.

◆ Large number of transitions leads to accumulation of dilution cullet.

Impact of a large number of transitions on the cullet inventorylevels:◆ a large amount of dilution cullet is produced, which leads to:

◆ violation of storage capacity◆ sell cullet with a penalty in the profit

◆ Schedule considers an unlimited supply of cullet◆ short runs deplete cullet inventory◆ violation of safety stocks of cullet

Scheduling without cullet may lead to infeasible cullet lev els


Full space MILP models

Detailed scheduling: slot based continuous time model◆ detailed timing of the schedule◆ slot based continuous time representation◆ variable length slots◆ one product per slot◆ products may take more than one slot◆ fixed number of slots per time period

Planning model: traveling salesman sequencebased model◆ Relaxation of the scheduling model◆ Sequencing constraints based on TSP

constraintsGeneric new features integrated in both models:◆ aggregation of products for changeovers with

no transition time◆ transitions across due dates◆ length of the production run and minimum run

lengths across due dates

Specific features:◆ cullet management


Cullet model

◆ Mass balance to the process (good and dilution cullet)◆ The consumption of cullet is a nonlinear function of the production length

◆ δ-form piecewise linear formulation (PLF) to approximate the cullet consumption

◆ The cullet consumption profile has 2 operating options for the total, good, and dilution cullet,(tc1,dc1,gc1) or (tc2,dc2,gc2)2

6

4

Z1i ,l ,t

tci ,l ,t = tc1i ,l ,t

dci ,l ,t = dc1i ,l ,t

3

7

5∨

2

6

4

¬Z1i ,l ,t

tci ,l ,t = tc2i ,l ,t

dci ,l ,t = dc2i ,l ,t

3

7

5∀i ∈ IM, l ∈ LM, t ∈ T

◆ Mass balance to the inventory (good, and dilution cullet) on a slot basis in the scheduling modeland time period basis in the planning model

◆ Cullet inventory constraints

◆ Option to sell cullet with a penalty


MILP model

Maximize the profit:

Profit = Sales - Inventory costs - Operating costs - Transition costs -- Product safety stock penalties - Cullet safety stock penalties -- Sell cullet penalty

Subject to:◆ Sequencing constraints◆ Product operating conditions◆ Product inventory balances◆ Product safety stocks and maximum capacity◆ Cullet operating conditions◆ Cullet inventory balances◆ Cullet safety stocks and maximum capacity

For a time horizon of 18 months, 25 products, and time periods of 1 month

Size of the planning, scheduling, and cullet model149,603 Equations167,270 Variables34,068 0-1 Variables


Solution approach

Rolling horizon algorithm (RHA) for the integrated planning, scheduling, and cullet model

◆ Time horizon: 24 months = 18 months schedule + 6 months planning◆ 6 months planning to provide feed back of future demand◆ 2 models are used: scheduling model (SC) and a planning model (PL)

The products not assigned in the first 6 months inPL are not considered in SC


Solution approach



1 - Fix the binary variables in the first 6 months2 - Extend the time horizon3 - Reduce the number of products in the SC model


Solution approach




Solution approach (cont.)

Rolling horizon algorithm with a hierarchical decomposition

◆ Assumptions

1. Priority to first satisfy the demand of the products◆ define a plan with the assignment of products to time periods

2. After defining the plan adapt the cullet supply and demand

3. Larger penalty for violations of the safety stocks of the products than the violation of the safetystocks of cullet




◆ Assumptions




Strategy to reduce the number of products

Goal: Provide flexibility to the products tomove to the next time period, when the culletmodel is added to the planning model




◆ Assumptions







◆ Assumptions







Remarks:

◆ The major advantage of this approach is that the size of the models is reduced.

◆ The disadvantage is that does not guarantee the global optimal solution since there is no iterationbetween the model with planning and planning and cullet.

◆ If the planning model determines a large number of transitions (increases generation of cullet) themodel with planning and cullet can reduce the number of transitions.


Case studies

Given:◆ time horizon of 18 months + 6 months◆ 12 substrates, 12 coated products, 1 different thickness.◆ demand due at the end of the month◆ products data◆ operating conditions

Case studies◆ Case 1 - Scheduling without cullet

Rolling horizon algorithm with integrated model

◆ Case 2 - Scheduling with culletRolling horizon algorithm with integrated model

◆ Case 3 - Scheduling with culletRolling horizon algorithm with hierarchical decomposition

Models implemented in GAMS and solved with Cplex 12.1 in a machine running Linux using 4 threads Xeon CPU,1.86GHz, and 8GB of RAM


Case 1 - scheduling without cullet

Case 1

Profit 1st year = 36.72m.u.

Objective function = 50.93m.u

# Transition days = 111.3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

JanFebMarApr

MayJunJul

AugSepOctNovDec

JanpFebpMarpAprpMaypJunp

Day

Mon

th

S−TR SG SB G

G B6

SG SB

B7 SB

ZB OG SB

SB C4−TR SG

SG TQ

VG SB SG

SG SB ZB AL

AL SG SB ZB

ZB SB SG

VG SG SB

SB B6 S−TR

VG SG B7

B7 AL SB

TQ ZB SG

AL SB

S−TR AL

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

Tot

al in

vent

ory

Time (days)

Case 1

Large number of transitions supported byunlimited cullet supply and storage. It is

infeasible.


Case 2 - scheduling with cullet, RHA integrated

Case 2


Objective function = -59,849.04m.u.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

JanFebMarApr

MayJunJul

AugSepOctNovDec


Day

Mon

th

B7 SG G

OG SB S−TR

S−TR ZB SG

B6 SB

SB SG

TQ SB

SB C4−TR

C4−TR VG SG

SG AL

SG B7

OG VG

SB

SG

SG AL TQ

AL

VG

B7 AL

AL S−TR B6

0 100 200 300 400 500 600 7000.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tot

al in

vent

ory

Time (days)

Case 1Case 2

Number of transitions decreases by consideringthe cullet balances. Solution has a large

optimality gap.


Case 3 - scheduling with cullet, RHA hierarchical

Case 3


Objective function = -24,705.64m.u.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

JanFebMarApr

MayJunJul

AugSepOctNovDec


Day

Mon

th

SB S−TR SG G

G ZB SG

SG SB

B7 B6 SG

SG SB

VG OG SG

SG C4−TR

TQ AL

SB SG

ZB

VG SB

SB SG

SG SB B7

B7 SG

B7

S−TR AL

AL

TQ

0 100 200 300 400 500 600 7000.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tot

al in

vent

ory

Time (days)

Case 1Case 2Case 3

Negative objective function is due toviolations of the safety stocks of cullet.


Profit, sales, and costs

Case 1 Case 2 Case 3

Profit (m.u.) 36.718 35.124 34.851

Sales (m.u.) 80.328 80.328 80.328

Inventory (m.u.) 7.326 7.854 7.846

Operating (m.u.) 28.904 31.370 32.277

Transitions (m.u.) 7.380 5.980 5.353

# Transitions: 28.0 22.0 22.0

# Transition days: 111.3 90.2 80.8

Amount below min. (ton): 0.0 298 0.9

Total backlog (ton): 0.0 10 0.0

Amount above max capacity (ton): 6004 4870 5085

In Case 2 and 3 number of changeovers decreases and the trade-off between operating costs and transition costs decreases theprofit.

All $ values have been normalized.


Computational resultsCase 1

Variables

SP # Eq. Total Binary Slots Gap (%) Profit (m.u.) CPU (s)

1 9,913 11,749 5,868 - 4.7 36.80 2443

2 7,425 7,995 4,140 19 3.4 36.61 281

3 12,778 14,263 7,618 19 4.9 42.59 1251

4 8,852 9,178 4,226 19 4.9 42.29 237

5 14,181 15,354 7,643 40 5.0 51.09 751

6 10,110 10,166 4,208 40 4.6 50.93 226

Minimum optimality gap and maximum CPU time: Gap = 5%, CPU = 3,600s. SP - Subproblem.



Variables


1 16,877 20,090 6,060 - 99.3 -28,906.79 3600

2 21,777 22,649 4,322 19 4.4 -31,572.01 890

3 30,188 30,795 7,871 19 0.4 -31,441.92 3042

4 29,349 26,727 4,177 34 2.1 -46,019.93 616

5 38,204 35,909 7,925 34 2.8 -46,819.17 2651

6 38,348 31,327 4,374 50 3.7 -59,849.04 739




Variables


1 16,877 20,090 6,060 - 99.3 -28,906.79 3600

2 21,777 22,649 4,322 19 4.4 -31,572.01 890

3 30,188 30,795 7,871 19 0.4 -31,441.92 3042

4 29,349 26,727 4,177 34 2.1 -46,019.93 616

5 38,204 35,909 7,925 34 2.8 -46,819.17 2651

6 38,348 31,327 4,374 50 3.7 -59,849.04 739

Case 3

Variables


1 9,913 11,749 5,868 - 4.9 36.74 622

2 8,805 9,652 1,141 - 15.3 -3,057.82 3600

3 17,071 17,081 1,726 18 0.0 -5,623.78 92

4 22,166 20,623 5,127 18 0.1 -5,608.68 21

5 25,839 20,917 1,538 35 4.3 -16,813.45 110

6 30,509 24,572 4,904 35 1.2 -15,557.71 12

7 33,892 27,423 4,004 49 1.1 -24,705.64 65




Variables


1 16,877 20,090 6,060 - 99.3 -28,906.79 3600

2 21,777 22,649 4,322 19 4.4 -31,572.01 890

3 30,188 30,795 7,871 19 0.4 -31,441.92 3042

4 29,349 26,727 4,177 34 2.1 -46,019.93 616

5 38,204 35,909 7,925 34 2.8 -46,819.17 2651

6 38,348 31,327 4,374 50 3.7 -59,849.04 739

Case 3

Variables


1 9,913 11,749 5,868 - 4.9 36.74 622

2 8,805 9,652 1,141 - 15.3 -3,057.82 3600

3 17,071 17,081 1,726 18 0.0 -5,623.78 92

4 22,166 20,623 5,127 18 0.1 -5,608.68 21

5 25,839 20,917 1,538 35 4.3 -16,813.45 110

6 30,509 24,572 4,904 35 1.2 -15,557.71 12

7 33,892 27,423 4,004 49 1.1 -24,705.64 65


The performance of the rolling horizon algorithm in Case3 is due to the reduced size of the MILPs .

The hierarchical decomposition performs better than the in tegrated ap-proach . However, global optimality of the solution is not guarantee d.


Implementation of the MILP model

From model development to decision support

◆ GAMS model integrated with Excel as a tool tosupport decisions

◆ GAMS models require an interface to input data anddisplay results.

◆ GDX and ASCII files are used to store input data andresults.

◆ Future users are familiar with data treatment in Excel.◆ Excel has a myriad of userform utilities to build a

front-end.


User friendly interface to handle data


Concluding remarks

Conclusions

◆ Scheduling without cullet management does not capture the global feasibility of the processoperation.

◆ The results show that integrating the planning and scheduling with the cullet management:

◆ increases substantially the size and complexity of the models used

◆ it is more difficult for the solver to close the integrality gap.

◆ The rolling horizon algorithm with a hierarchical approach does not guarantee global optimalsolutions, but provides better solutions faster than the integrated approach.

Future Work

◆ Improve the model to provide tighter linear relaxations

◆ Include another production line in the model

◆ produces some of the current products

◆ integrated through the cullet management and inventory space

Documents

Carnegie Mellon University - Planning and …egon.cheme.cmu.edu/ewo/docs/PPGrlima_ewo_march_2010.pdfCarnegie Mellon University EWO meeting, March 2010 - p. 2 Project description Objectives: