Production Planning for Batch Operationsegon.cheme.cmu.edu/ewocp/docs/EWODowMarch1407.pdf · Production Planning for Batch Operations ... Proposed Planning Model RP Mass Balance and

Production Planning for Batch OperationsProduction Planning for Batch Operations

Müge Erdirik Doğan and Ignacio E. GrossmannCarnegie Mellon University

John WassickESMD Process Optimization

The Dow Chemical Company

Enterprise-wide Optimization ProjectMarch 2007

2

Motivation and IntroductionMotivation and Introduction

Goal: To develop production planning modelsDetermine the available production capacity accuratelyAccounting for sequence-dependent changeovers

Real world problem:Specialty Chemicals and Plastics business within Dow Chemical

Business challengesIntroduction of new productsCost pressures

Flexibility increases the complexity in the planning processAccurate production capability offers significant competitive advantage

3

Problem StatementProblem Statement

Production Site:Raw material availability and Raw material costsStorage tanks with associated capacity Reactors:

Materials it can produceBatch sizes (lbs) for each material it can produceOperating costs ($/hr) for each materialSequence dependent change-over times(hrs per transition for each material pair)Time the reactor is available during a given month (hrs)

Customers:Monthly forecasted demands for desired productsPrice paid for each product

Materials:Raw materials, Intermediates, Finished productsUnit ratios (lbs of needed material per lb of material

produced)

F1

F2

F3

F4

Reaction 1 A

Reaction 2 B

Reaction 3 C

INTERMEDIATESTORAGE

STORAGE

STORAGE

STORAGE

week 1 week 2 week t

due date due date due date

week 1 week 2 week t

due date due date due date

4

Problem StatementProblem Statement

Production quantities Inventory levels Number of batches of each product Assignments of products to available processing equipmentSequence of production in each processing equipment

OBJECTIVE:OBJECTIVE:

To Maximize Profit.Profit = Sales – CostsCosts=Operating Costs – Inventory Costs –Transition Costs

DETERMINE THE PRODUCTION PLAN:DETERMINE THE PRODUCTION PLAN:

5

Proposed MILP Planning ModelsProposed MILP Planning Models

OPTION A. (Relaxed Planning Model-RP)Constraints that underestimate the sequence-dependent changeover times

=> Weak upper boundsOPTION B. (RP*)

Simplification of Relaxed Planning ModelTightening transition constraints are neglected

OPTION C. (Detailed Planning Model-DP)Sequencing constraints for accounting for changeovers rigorously

=> Tight upper boundsOPTION D. (Relaxed Detailed Planning Model-DP*)

Relaxation of DPNumber of batches are treated as continuous variables

OPTION E. (Rolling Horizon Approach- RH)Forward rolling horizon algorithm

6

SimplificationSimplification

Intermediate storage tank and the dedicated storage tank aggregated into a single tank Aggregate mass balances for the intermediates

F1

F2

F3

F4

Reaction 1 A

Reaction 2 B

Reaction 3 C

INTERMEDIATESTORAGE

STORAGE

STORAGE

STORAGE

STORAGEF1

F2

F3

F4

Reaction 1 A

Reaction 2 B

Reaction 3 C

STORAGE

STORAGE

7

Generic Form of the Relaxed Planning Model (RP)Generic Form of the Relaxed Planning Model (RP)-- Option AOption A

RELAXED PLANNING MODEL (RP)

Mass Balances on State Nodes

Time Balance Constraints on Equipment

Objective Function

PSFPFPSETI∈I∈

Reactor RReactor R

Available time for R

Product 1 TransitiontimeProduct 2

Transitiontime ……………………..

,j tP,j tS

, ,i m tFP , ,i m tFPSETOI∈

jI CS∈

8

Key Variables for the Model (RP)Key Variables for the Model (RP)

, ,i m tYP :the assignment of products to units at each time period

imtNB :number of each batches of each product on each unit at each period

imtFP :amount of material processed by each task

Reactor 1 or Reactor 2 or Reactor 3Products: A, B, C, D, E, F

T1 T2 T3

F B F C DR3 E

R2 F B F C DB B

D B BR1 AA A B E

, 1, 1

, 1, 1

1

3A reactor time

A reactor time

YP

NB

=

=, 2, 2

, 2, 2

1

2B reactor time

B reactor time

YP

NB

=

=

9

Proposed Planning Model RPProposed Planning Model RP

Mass Balance and Assignment Constraints:

imt imt imtFP Bound YP≤ ⋅

imtimimt YPQFP ⋅≥

•Bound on production levels of product i on unit m at time period t.

•Sets production level to zero if product i is not assigned onunit m at time period t.

Largest number that the task can be repeated

timt im

im

HBound QBT

= ⋅

Maximum capacity

Maximum lbs of product i produced on unit m at time period t if product i is assigned throughout the time period

, , , , ,i m t i m t i mNB FP Q= Number of batches of product i in unit m at time t (INTEGER)

∑ ∑∑ ∑∈

−∈∈ ∈

−++=+j ij i CSi

jtjtMTm

imtjiPSi

jtMTm

imtjijt INVINVFPSFPP 1ρρ

purchases production sales consumption change in inventory

Mass balance on each state node:

10

Transitions and Time Balance Constraints for RPTransitions and Time Balance Constraints for RPLower bounds for changeoversSequencing constraints are neglected Introduce a minimum transition time for each assigned product

{ }, ''i i ii iTR Min τ

≠=

Parameter:

,A mBT ATR CTR BTR

Period t

A CTR BTRA C B

Period tPeriod t

,C mBT ,B mBT ,A mBT,C mBT CTRATR

Period t+1

CTRATRCTRC ATR

Period t+1Period t+1

, , , , , ,i m t i m i i m t ti i

NB BT TR YP H m t⋅ + ⋅ ≤ ∀∑ ∑

,A mBT

Period t

ATRATRA CTRCTRC BTRBB

Period t

,C mBT ,B mBT ,A mBT ATRAACTRCTRCBTRBB

Period t +1

,C mBT,B mBT

Over estimation of changeoversCan not model transitions across adjacent periods

, , , , , , ,i m t i m i i m t m t t

i iNB BT TR YP U H m t⋅ + ⋅ − ≤ ∀∑ ∑

11

Transitions and Time Balance Constraints for RPTransitions and Time Balance Constraints for RP

, , , , , ,m t i m i m t mU TR YP i I m t≥ ⋅ ∀ ∈

, , , ,m

m t i i m ti I

U TR YP m t∈

≤ ⋅ ∀∑

, , ,( )p inv operjt jt jt jt it imt i i m t m t

j t j t i m t t m iZ cp S c INV c FP TRC YP UT= ⋅ − ⋅ − ⋅ − ⋅ +∑∑ ∑∑ ∑∑∑ ∑∑ ∑

Sales Inventorycosts

Transition costsVariable operating costs

Maximize PROFIT:

Transition Costs:

{ }, '' i ii transi iTRC Min C

≠=

, , ,( )i i m t m tt m i

Transition Cost TRC YP UT= ⋅ −∑∑ ∑

{ }, , ,m

m t i mi IU Max TR m t

∈≤ ∀

where :,m tU

(*)

(*) Redundant, tightens the formulation, neglecting may result in overestimation of the available production time.

where:

, , , , , ,m t i m i m t mUT TRC YP i I m t≥ ⋅ ∀ ∈

{ }, , ,m

m t i mi IUT Max TRC m t

∈≤ ∀

, , , ,( ) ,m

m t i m i m ti I

UT TRC YP m t∈

≤ ⋅ ∀∑ (**)

(**) Redundant, tightens the formulation, neglecting may result in underestimation of the transition cost.

12

Generic Form of Model (RP*)Generic Form of Model (RP*)-- Option BOption B

PLANNING MODEL (RP*)

RP* same as RP but constraints (*) and (**) are neglected.

Provides a valid but a weaker upper bound compared to RP.

It can lead to overestimation of the available production time and underestimation of transition costs.

13

Generic Form of the Detailed Planning Model (DP)Generic Form of the Detailed Planning Model (DP)-- Option COption C

DETAILED PLANNING MODEL (DP)

Mass Balances on State Nodes

Sequencing ConstraintsSequence dependent changeovers determinedDetailed timings of operations neglected

Time Balance Constraints on Equipment

Objective Function

PSFPFPSETOI∈I∈

Reactor RReactor R

Available time for R

Product 1 TransitiontimeProduct 2

Transitiontime ……………………..

,j tP,j tS

, ,i m tFP , ,i m tFPSETOI∈

jI CS∈

14

Sequencing Constraints for DPSequencing Constraints for DP

Sequence dependent changeovers:Sequence dependent changeovers within each time period:

1. Generate a cyclic schedule where total transition time is minimized.KEY VARIABLE:

mtiiZP ' :becomes 1 if product i is after product i’ on unit m at time period t, zero otherwise

P1, P2, P3, P4, P5 P1

P2

P3

ZP P1, P2, M, T = 1

ZP P2, P3, M, T = 1

mtiiZZP ' :becomes 1 if the link between products i and i’ is to be broken, zero otherwise KEY VARIABLE:

2. Break the cycle at the pair with the maximum transition time to obtain the sequence.

P1

P2

P3P4

P4

?ZZP P4, P3, M, T

P4

P4P5

15

Changeovers within each period for DPChangeovers within each period for DP

P1

P2

P3P4

P4

P2, P3, P4, P5, P1 ZZP P1, P2, M, T = 1

P3, P4, P5, P1, P2 ZZP P2, P3, M, T = 1

P4, P5, P1, P2, P3 ZZP P3, P4, M, T = 1

P5, P1, P2, P3, P4 ZZP P4, P5, M, T = 1

P1, P2, P3, P4, P5 ZZP P5, P1, M, T = 1

P1

P2

P3P4

P4

P2, P3, P4, P5, P1 ZZP P1, P2, M, T = 1

P3, P4, P5, P1, P2 ZZP P2, P3, M, T = 1

P4, P5, P1, P2, P3 ZZP P3, P4, M, T = 1

P5, P1, P2, P3, P4 ZZP P4, P5, M, T = 1

P1, P2, P3, P4, P5 ZZP P5, P1, M, T = 1

According to the location of the link to be broken:

The sequence with the minimum total transition time is the optimal sequence within time period t.

''

, ,imt ii mti

YP ZP i m t= ∀∑' ' ', ,i mt ii mt

i

YP ZP i m t= ∀∑

''

1 ,ii mti i

ZZP m t= ∀∑∑' ' , ', ,ii mt ii mtZZP ZP i i m t≤ ∀

Generate the cycle and break the cycle to find theoptimum sequence where transition times are minimized.

Having determining the sequence, we can determine the total transition time within each week.

' ' , ,[ ]i iimt i mt iimtYP YP ZP i m t≠¬ ∀∧ ∧ ⇔

, , , , ,imt i i m tYP ZP i m t≥ ∀

, , , ', , 1 , ' , ,i i m t i m tZP YP i i i m t+ ≤ ∀ ≠

, , , , , ', ,'

, ,i i m t i m t i m ti i

ZP YP YP i m t≠

≥ − ∀∑

' ' , ,[ ]i iimt i mt iimtYP YP ZP i m t≠¬ ∀∧ ∧ ⇔

, , , , ,imt i i m tYP ZP i m t≥ ∀

, , , ', , 1 , ' , ,i i m t i m tZP YP i i i m t+ ≤ ∀ ≠

, , , , , ', ,'

, ,i i m t i m t i m ti i

ZP YP YP i m t≠

≥ − ∀∑

16

Changeovers within each period for DPChangeovers within each period for DP

, , ' , ', , , ' , ', ,' '

,m t i i i i m t i i i i m ti i i i

TRNP ZP ZZP m tτ τ= ⋅ − ⋅ ∀∑∑ ∑∑

P4 P5 P1 P2 P3

4, 5P Pτ5, 1P Pτ 1, 2P Pτ 2, 3P Pτ

3, 4P Pτ

P1

P2

P3P4

P5

ZZP P4, P3, M, T =1

1) generate the cycle

2) break the cycle to obtain the sequence

Total transition time within period t on unit m

, 4, 5 5, 1 1, 2 2, 3 3, 4 3, 4m t P P P P P P P P P P P PTRNP τ τ τ τ τ τ= + + + + −

Transition time required to change the operation from P1 to P2

17

Changeovers across adjacent periods for DPChangeovers across adjacent periods for DPSequence dependent changeovers across adjacent time periods:

P1

P2

P3P4

P5

ZZP P1, P2, M, T = 1

Tail

Head P2 P3 P4 P5 P1

First element

Last element

XLi, m, t

P1t P2t P4t P1t+1 P2t+1

XFi’, m, t+1

T T+1

, ', , , , ', , 1 1 , ', ,i i m t i m t i m tZZZ XL XF i i m t+≥ + − ∀

Transitions across adjacent weeks

18

Time Balance and Objective Function for DPTime Balance and Objective Function for DP

The time balance constraint:

, , , , , ', , ''

,i m t i m m t i i m t ii ti i i

NB BT TRNP ZZZ H m tτ⋅ + + ⋅ ≤ ∀∑ ∑∑

summation of batch timesof assigned products to unit mat period t.

total transition timewithin time period t

transition time betweenperiod t and period t+1

total available timefor unit m

, ' , ', , , ' , ', , , ' , ', ,' ' '

p inv oper trans trans transjt jt jt jt it imt i i i i m t i i i i m t i i i i m t

j t j t i m t i i m t i i m t i i m tZ cp S c INV c FP c ZP c ZZP c ZZZ= ⋅ − ⋅ − ⋅ − ⋅ + ⋅ − ⋅∑∑ ∑∑ ∑∑∑ ∑∑∑∑ ∑∑∑∑ ∑∑∑∑

Sales Inventorycosts

Transition costsVariable operating costs

Maximize PROFIT:

Objective Function:

19

Generic Form of Model (DP*)Generic Form of Model (DP*)-- Option DOption D

RELAXED DETAILED PLANNING MODEL (DP*)

DP* same as DP but number of batches are treated as continuous variables.

Provides a valid upper bound on the profit.

Significantly reduces the computational expense.

20

Rolling Horizon Approach Rolling Horizon Approach –– Option EOption E

The proposed planning models may still be to very expensive to solve. Sequence of sub-problems that are solved recursively.Provides a lower bound on the profit.

DP RPDP

Week 1 Week 2

DP RPDP

Week 1 Week 2

DPDP

Week 3 Week 4fixed

DP RPDP

Week 1 Week 2

DPDP

Week 3 Week 4fixed

Week 5 Week 6

DPDP

fixed

DP RPDPWeek 1 Week 2 Week 3

DP

DP RPDPWeek 1 Week 2

DPDPWeek 3 Week 4

fixed

DP DPWeek 5 Week 6

DP DPWeek 1 Week 2

DPDPWeek 3 Week 4 Week 5 Week 6

DPDP DP

fixed fixed

The detailed planning period (DP) moves as the model is solved in time.Future planning periods include only underestimations for transition times (RP*).In each iteration we fix the binary variables for assignment and sequencing variables.

21

REMARKSREMARKS

Relaxed Planning Model (RP) is adequate if:Demand rates are lowTransition times show low variance

RP could lead to significant overestimation of the available production capacity if the above conditions are not true.

Detailed Planning Model (DP) very powerful:Since the sequencing constraints are explicitly accounted for, it yields very tight upper bounds.In the absence of subcycles and for single stage production, it produces the identical solution as a detailed scheduling model would.Among all the instances we have solved so far, only one exhibited a subcycle.

Trade-off between the extent of scheduling decisions incorporated and the size and the computational effort of the resulting problem.

ExamplesExamples

23

EXAMPLE 1 EXAMPLE 1 -- 8 Products, 3 Reactors8 Products, 3 Reactors

Reaction 1

Reaction 2

Reaction 3

R2

A

B

C

Reaction 4 D

Reaction 5 E

Reaction 6

Reaction 7

Reaction 8

F

G

H

R1

R3

•8 Products, A,B,C,D,E,F,G,H•All produced in a single stage.•3 Reactors, R1,R2,R3•End time of the week is defined as due dates•Demands are lower bounds

Determine the plan for 8 products, 3 reactors plant so as to maximize profit.

24


week 1 week 6week 5week 4week 3week 2R1

week 1 week 6week 5week 4week 3week 2

R2


R3

C A B

(2) (2) (6)

B C

(11) (2)

C A

(4) (4)

A B

(4) (8)

B

(16)

B

(16)

D F

(3) (4)

F E

(5) (4)

E D

(8) (2)

D E

(6) (2)

E

(11)

E

(11)

G E H

(5) (3) (2)

H

(8)

H E

(2) (8)

E

(11)

E G

(8) (4)

G

(16)

a)



R2


R3

C A B

(3) (2) (3.5)

C A

(5) (2.4)

A B

(5.6) (5.7)

B

(16.8)

B

(16.8)

D F

(3) (4.8)

F E

(4.2) (5.1)

E D

(8) (2)

D E

(6) (2.4)

E

(11.2)

E

(11.2)

H E G

(2) (3.6) (3)

H E

(3.1) (6.5)

E

(11.2)

E

(11.2)

B

(16.3)

G H

(4) (4.9)

E G

(8.8) (3)

b)



R2


R3

C A B

(2) (2) (6)

B C

(11) (2)

C A

(4) (4)

A B

(4) (8)

B

(16)

B

(16)

D F

(3) (4)

F E

(5) (4)

E D

(8) (2)

D E

(6) (2)

E

(11)

E

(11)

G E H

(5) (3) (2)

H

(8)

H E

(2) (8)

E

(11)

E G

(8) (4)

G

(16)


week 1 week 6week 5week 4week 3week 2week 1 week 6week 5week 4week 3week 2R1


R2week 1 week 6week 5week 4week 3week 2week 1 week 6week 5week 4week 3week 2

R2



R3

C A B

(2) (2) (6)

C A B

(2) (2) (6)

B C

(11) (2)

B C

(11) (2)

C A

(4) (4)

C A

(4) (4)

A B

(4) (8)

A B

(4) (8)

B

(16)

B

(16)

B

(16)

B

(16)

D F

(3) (4)

D F

(3) (4)

F E

(5) (4)

F E

(5) (4)

E D

(8) (2)

E D

(8) (2)

D E

(6) (2)

D E

(6) (2)

E

(11)

E

(11)

E

(11)

E

(11)

G E H

(5) (3) (2)

G E H

(5) (3) (2)

H

(8)

H

(8)

H E

(2) (8)

H E

(2) (8)

E

(11)

E

(11)

E G

(8) (4)

E G

(8) (4)

G

(16)

G

(16)

a)



R2


R3

C A B

(3) (2) (3.5)

C A

(5) (2.4)

A B

(5.6) (5.7)

B

(16.8)

B

(16.8)

D F

(3) (4.8)

F E

(4.2) (5.1)

E D

(8) (2)

D E

(6) (2.4)

E

(11.2)

E

(11.2)

H E G

(2) (3.6) (3)

H E

(3.1) (6.5)

E

(11.2)

E

(11.2)

B

(16.3)

G H

(4) (4.9)

E G

(8.8) (3)


week 1 week 6week 5week 4week 3week 2week 1 week 6week 5week 4week 3week 2R1



R2



R3

C A B

(3) (2) (3.5)

C A B

(3) (2) (3.5)

C A

(5) (2.4)

C A

(5) (2.4)

A B

(5.6) (5.7)

A B

(5.6) (5.7)

B

(16.8)

B

(16.8)

B

(16.8)

B

(16.8)

D F

(3) (4.8)

D F

(3) (4.8)

F E

(4.2) (5.1)

F E

(4.2) (5.1)

E D

(8) (2)

E D

(8) (2)

D E

(6) (2.4)

D E

(6) (2.4)

E

(11.2)

E

(11.2)

E

(11.2)

E

(11.2)

H E G

(2) (3.6) (3)

H E G

(2) (3.6) (3)

H E

(3.1) (6.5)

H E

(3.1) (6.5)

E

(11.2)

E

(11.2)

E

(11.2)

E

(11.2)

B

(16.3)

B

(16.3)

G H

(4) (4.9)

G H

(4) (4.9)

E G

(8.8) (3)

E G

(8.8) (3)

b)

For a planning horizon of 6 weeks:

Schedule and Number ofBatches obtained by DPExact schedule!

Schedule and Number ofBatches obtained by DP*

methodnumber of

binary variablesnumber of

continuous variablesnumber of equations

time(CPU s)

solution ($)

detailed planning (DP) 864 1327 1483 1667 11,819detailed planning (DP*) 792 1327 1483 96 12,211relaxed planning (RP) 144 349 553 1.75 13,460rolling horizon (RH) 624 1,291 1,483 322 11,377

25


0

10,000,000

20,000,000

30,000,000

40,000,000

50,000,000

60,000,000

6 Weeks 12 Weeks 18 Weeks 24 Weeks

($)

DP*

RP

RH

1

10

100

1,000

10,000

100,000


CPUs

DP*

RP

RH

Comparison of Models for Planning Horizons of 6 to 24 Weeks for 5% Optimality Tolerance

26

EXAMPLE 2 EXAMPLE 2 -- 15 Products, 6 Reactors, 48 Weeks15 Products, 6 Reactors, 48 Weeks

R2

Reaction 1 A

Reaction 2 B

Reaction 3 C

Reaction 4 D

Reaction 5 E

Reaction 6 F

Reaction 7 G

Reaction 8 H

R1

R3

Reaction 9 J

Reaction 10 K

Reaction 11 L

Reaction 12 M

Reaction 13 N

Reaction 14 O

Reaction 15 P

R4

R5

R6

• 15 Products, A,B,C,D,E,F,G,H,J,K,L,M,N,O,P• B, G and N are produced in 2 stages.• 6 Reactors, R1,R2,R3,R4,R5,R6• End time of the week is defined as due dates• Demands are lower bounds

Determine the plan for 15 products, 6 reactors plant so as to maximize profit.

27

EXAMPLE 2 EXAMPLE 2 -- 15 Products, 6 Reactors, 48 Weeks15 Products, 6 Reactors, 48 Weeks

0

50,000,000

100,000,000

150,000,000

200,000,000

250,000,000


($)

(RH)

(RP)

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000


(CPUs)

(RH)

(RP)

a) b)0

50,000,000

100,000,000

150,000,000

200,000,000

250,000,000


($)

(RH)

(RP)

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000


(CPUs)

(RH)

(RP)

a) b)

For a planning horizon of 48 weeks for 6% optimality tolerance :

Variation of results from 6 to 48 weeks for 6% optimality tolerance :

method

number of binary

variables

number of continuous variables

number of equations

time(CPU s)

solution ($)

relaxed planning (RP) 2,592 5,905 9,361 362 224,731,683rolling horizon (RH) 10,092 25,798 28,171 11,656 184,765,965rolling horizon (RH**) 1,950 25,798 28,171 4,554 182,169,267

28

29

Concluding Remarks and SummaryConcluding Remarks and Summary

Relaxed Planning Model (RP):Underestimates sequence-dependent changeover times and costs.Overestimates sales and profit.

Detailed Planning Model (DP): Explicitly accounts for scheduling via sequencing variables and constraints.Very accurate production plans

DP yields more realistic plans compared to RP but at the expense of increasing size and computational effort.

For large problems and long time horizons without giving up the solution quality:

Rolling Horizon Algorithm (RH): yields a lower bound on profitRelaxed Detailed Planning Model (DP*): yields an upper bound on profit

Documents

Production Planning for Batch Operationsegon.cheme.cmu.edu/ewocp/docs/EWODowMarch1407.pdf · Production Planning for Batch Operations ... Proposed Planning Model RP Mass Balance and