1.
 Optimal Scheduling of Non-Sequential Multipurpose Batch Processeswith Various Intermediate Storage Policies


       

 

^† *

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(1999 10 25 , 2000 4 4 )

Optimal Scheduling of Non-Sequential Multipurpose Batch Processes with Various Intermediate Storage Policies

Sang Beom Kim, Ho-Kyung Lee, In-Beum Lee^† and Euy Soo Lee*

Automation Research Center, Dept. of Chemical Engineering, POSTECH,

*Department of Chemical Engineering, Dongguk University, (Received 25 October 1999; accepted 4 April 2000)



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Abstract−In this paper, we present mathematical models for optimal scheduling of non-sequential multipurpose batch pro- cesses under various intermediate storage policies. Compared with multiproduct processes, in non-sequential multipurpose batch processes, the production routes of products may be different from one another and may be backward direction. Conse- quently, in order to reduce idle time of units and to raise the efficiency of process, we have to make operation sequences of products in each unit differently by considering processing route of each product with a given intermediate storage policy. The basic concept of our approach for these problems is as follows. Firstly, we represented the starting and finishing time of a task in each unit with two coordinates for a given storage policy. One is based on products, and the other is based on sequences.

Then, we matched the variables used in the two coordinates into one with binary variables and logical constraints. We formu- lated these problems as MILP(Mixed Integer Linear Programming) models and apply them to three examples to show the effectiveness of the model.

Key words: Scheduling, Multipurpose Batch Process, Intermediate Storage Policies, MILP

†E-mail: iblee@postech.ac.kr

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2.

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Table 1R b @ k DhU UIS NwLM ;<›"+ [, Y

→B→C)M  c3\# VF ãä (Gantt) Wä -âåLU, Fig. 2(b) # UIS Nw 4Q@yz

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Fig. 1. Illustrative example.

Table 1. Data for illustrative example Unit

Product

Unit 1 Unit 2 Unit 3

Processing times

A 3 8 4

B 7 5 2

C 6 4 7

Fig. 2. (a) Schedule for the same production sequence(A→→→→B→→→C) in all→ units, and (b) Optimal schedule for Fig. 1 under UIS policy.

 39 1 2001 2

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2-2.  

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Ò(|I_j|=|K_j|)

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(1)

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3-1. UIS 
     

UIS Nw VF# v /  ·# VFM  ®{

@ ÷i  "ë ®{ M 9D+ [ "ë ® { D "x  c3=+ !# GP+ =¶Pt ¨ ®{

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ëR b".

[ @ Ö kR yz k  # "ëR b £{D 1
".

(3)

(4)

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 i  -õh n “0” \dM Bf kóôM c3\#

 @k± -â' ›".  (4)#   ÓÝ Ö@

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ó  7pLM @ ÖR yzk ¨ •m.

(5) (6)

 (5)# ®{ j  i @ Ö kR yzk  £{

 -âU,  (6) (j_)_i#  i  VMŸ j®{ "ë ®

 @ Ö  !ë -âò".

  (3)-(6) ù _  7pLM ¨› @k Ê

[

(7) (8)

 (7)R (8)  iD  j kóôM c3\# VF ± êµ

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r !# q   UM . y|\9  œ û gh Ž

ÖR yzk µLM ¨\9 !7 [C  (7)R (8) Ó

K. e9  G 9> =-± /=È =#    !".

X_ikj

k

∑

X_ikj

k

∑

TFU_kj TSU_kj t_ijX_ikj

i

∑

+

=

TSU_kj≥TFU_{(k 1– )j} ∀, k∈K_j t_ijX_ikj

i

∑

TF_ij=TS_ij+t_ij i∀,j∈J_i TS_{i j( )−i}≥TF_ij ∀, j∈J_i

U 1 X( – _ikj)

– ≤TS_ij–TSU_kj≤U 1 X( – _ikj) i∀, j∈J_i, k∈K_j U 1 X( – _ikj)

– ≤TF_ij–TFU_kj≤U 1 X( – _ikj) i∀, j∈J_i, k∈K_j

SKª 4Q@yzk 4|P+ [ UIS Nw 5

j# 

 j D -G Ý\# @ -âU, k1 D év Ý

\# @ -âò".

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ûü j

ûü j

 (1)-(8) ûü

8 Y¯ SKª_ ûü Ya ÓÔU, 0§ Ê_ 

» Ê ªÃ /=# ÑÒÓÔ{¦ CD ›".

3-2. ZW 
     

ZW Nw UIS NwR# ƒ3 (   Gk vD ·+,

 ‡ˆŸ  ®{ @ ÷i  h · °M "ë 

µLM ¨\‚  7p ¨ (6) "ëR b µ ü LM °' ›".

(9) Ü UIS Nw 5 Y¯  (6)  (9)M 5=È ZW Nw LM ;<\# "SK
  4K@A B Y¯ ›".

3-3. NIS 
     

NIS Nw VF# ( ®{   Gk  v v

D ·7 [C , ¨ ®{ @ ÷¶Pt "ë ®{ D

@G. VF # "ë ®{M 9Dh ¡=+ ¨ ®{ Gk 

 •  57=# NLM ;<›". OP, UIS- ZW Nw R# ƒ3 ( „"  ‡ˆŸ "ë ®{ @ Ÿ +=

¨ ®{ Ö@ Qz=+ "ë ®{ @LM 9 e.h,

ÂÈ ¨ ®{ 57 e.h B”± ".  8 û

ü 3=È "ëR b".

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@ Qz\¶Pt 57=# k   !7 [C "ëR b

 êµLM °' ›".

(10)

(11)

`, vD ·7 [C   @ yzkR "ë 

# 굂  (6) ZW N  (9)c µ ›".

 (  @ ‡ˆ ¨ ®{ @ŸR "ë ®{

@ Ÿ X×=# û ü -âÈ "ëR b".

(12)

 (12) ( j_)_iD g=# ®{# "Q
 R ƒ3 Y Z „" M "…7 [C iD °È (j_)it OP °' ›".

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 j @ yz k  ‡ˆŸ "ë ®{ (j_)i (k−1) óôM c3\#  @ yz k Å9” ª -âò". 

(12) VFt  “1”. VF ± <‰ g' \+, -õh

VF# q   UM . y|\9  œ <‰ gh Ž' ›

".

UIS Nw 5 Y¯  (3), (5)-(6)  (9)-(11)M °o+,

 (12) ¶=È NIS Nw 5 4K@A B Y¯ ›".

3-4. FIS 
    FIS

 j § Gk v j' , ®{  c3\# ±

v  !"+ D". OP, v ðª  j VF#

,  c3\+ i   ‡ˆŸ "ë c3 ®{ M

9D+ [ "ë ®{ D X9 !"È °M "ë M  9  !h±, "ë D @ =+ !"È ,  / Dì [ºh Gk v D/ ê OP Gk  vM •

v=]-, ,Œh ¡ VF# ¨  57”± ". -õh G k vD ·#  VF# Ü NIS NR I=' @›"

+ D".

XAW "SK  FIS NLM @\# VF vD !#

 j  iD @ yzö [ I9  !# 4Dh Ÿ 3

=È "ëR b".

•"ë  j+1 X9!+, vt X9!# VF

•"ë  j+1 X9!+, v# W!# VF

•"ë  j+1 @G+, v# X9!# VF

•"ë  j+1 @G+, vt W!# VF

8 4Dh VFG  óô_ a óô VF ®{ j @ yz

›  v ]h Ž+ °M "ë ®{ M 9  ! +, Ø óô VF# Gk v  v=' \U, óô VF

# ¨ ®{ 57=+ !"D "ë ®{ - v G év X

# !LM 9D' ›".

FIS Nw 5 @ yz ¨ [ "Q
 R "

SK
  D  W$ "Q
  # YZ

  ‡ˆ I=7 [C Ó"Ó#(First Input First Output)

3M ;<\#  a óô_ b Ÿ ø5 I9-h Ž#". "

 $=È Gk vD W !"È J% "ë ®{  (j_)it 

@ GdM ¨  j @ ÷È ®Aq v / D ì ê± +=È ›".  X "SK
  VF# j

[C a óô_ b Ÿ €  !". OP, v ðª

 ®{ @ yz k 8 4Dh Ÿ Ya K/ Dì  3K û üLM ¨”± ".

év  j § v j'P [ v Gk  v

ê B=# Ê SX_ikj' 5 û ü "ëR b".

(13)

 (13) v ðª  j kóôM c3D ÷i  i#

v j' vö t !+ & t !ë -âU, vD 

$ =- ' v  ·ë g=7t ".

 v ðª  j c3D ÷i  v j'_ "

ë  (j_)i / ê OP ¨  @ ÷(  !#h 57 e.h ê B=# û ü "ëR b".

(14)

(15)

(16) LFT TFU

kjj

≥ EST TSU_k

1j

≤

TS_{i j}

( )−i=TF_ij ∀, j∈J_i

TFU_kj TSU_kj t_ijX_ikj

i

∑

≥ +

TF_ij≥TS_ij+t_ij ∀, j∈J_i

TF_ij TFU_{(k 1– )j}

( )−i U 1 X_{ik j}

( )−i

( – )

– ∀j, j∈J_i, k∈K_{j 1+}

≥

X_{ik j}

( )−i

X_ikj≥SX_ikj′ i∈I_j, k∈K_j

TF_ij≥TFU_{(k 1– )j′}–U 1 SX( – _ikj′) i∈ _j, kI∈K_j

TF_ij′ TFU_{(k′–1)( )j−i} U 2 S

k

∑

– –X_{ij′( )j−i}

 

  i∈I_j′,k′∈K_{( )j−i}

≥ –

U 1 S

k

∑

 – 

  TS_ij′–TF_ij U 1 S

k

∑

 – 

 

≤

≤ i∈I_j′

–

 39 1 2001 2

(17)

 (14)# ®{ j c3D ÷i  Gk v v=+

=È év v\9 !  v ÷- vD X9 !9” ª

 g=U,  (15)# Gk v v\9 !  "ë ®{

M 9D+ [ "ë ®{  ÓÝ Ö@ ÷-” ª g

".  (16)  j c3D ÷i  iD v v\# V

}7 8 û ü".  (17) ®{ j c3›  v

 ]h Ž+ °M "ë ®{M 9 VF "ë ®{

ÓÝ Ö@ ÷- D X9 !9”ª D3)".

 Gk v v\# VF #  (14)-(16) û *LU,  (17)  œ û gh Ž' ›". JÈ, v v\h Ž+

°M "ë ®{M 9D# VF #  (14)-(16) y|\9  œ

<‰ gh Ž+  (17)± û *' ›".

œ VF # vD ·# "x nR I=' "ë 

D/ ê± +=# VFD \U, ±I YZ  v /

=h Ž' \È Bf NIS N VF_ I BR ' ›".

_ b "Q
  # YZ  (j_)i®{D b7 [C ¨ ®{  j @ ÷È ®Aq v / Dì  ê± +=È \‚h±, "SK
  # ( „" (j_)i

®{D M "…7 [C v_ "ë  (j_)i D/ ê 

 +” "# $  WD !".

 v ðª=h Ž# -õh ®{n 5# NIS Nw /\# ûü ,5M /=È ›".

4. $ %

 ,9 , BR •­". MILP C -#Ú / solver#

GAMS/OSLU, IBM RS/6000(Y¯ 350) š&./0 /=­".

4-1.  1

¿M  9»  5  Y¯ K/=­". 5Ÿ  Fig. 3 R bLU @k Úí# Table 2 -â‚". ParkR Jung[8]

=#Ú ,1 JÈ, ¼ C  Y¯ ¶ -D 4Q @

yzk 4| }# @A B  !"# $ "…".

X× 8 Fig. 4 # ¼ C  Y¯ / B UIS Nw 5 4K @ A_ 4Q@yzk ãä WäM -â

‚". ParkR Jung[8] ² @yzk “170”‚h± 

D -s".

4-2.  2

Fig. 5 -âi  4¿,  4¿M  9»  5  Y¯

 K/=­". @ k Úí# Table 3 e /²LU, ( v

 ;<Nw K/=  BR Table 4R 5 -â‚".

TF_ij TFU_(k′–1)j

( )−i U 1 S

k

∑

 + 

 i∈I_j′, k′ K_j

( )−i

∈

≥ –

Fig. 3. Schematic diagram for example 1.

Table 2. Data for example 1 Unit

Product

Unit 1 Unit 2 Unit 3

Processing times

P1 20 25 30

P2 15 10

P3 30 30

P4 20 15

P5 25 30

P6 35 30 20

Fig. 4. Result for UIS policy of example 1.

Table 5. Statistical result for example 2

0-1 variable Continuous variable Constraint Makespan CPU(sec)

UIS 52 58 196 59 0.43

ZW 52 58 196 71 0.43

NIS 52 58 317 63 0.85

FIS 68 73 361 60 1.24

Table 4. Optimal sequence in each unit for example 2

Unit 1 Unit 2 Unit 3 Unit 4

UIS B-A-C C-B-D C-D-A-B D-A-B

ZW B-C-A C-B-D C-D-B-A D-B-A

NIS A-B-C C-B-D C-A-D-B D-A-B

FIS B-A-C B-C-D C-D-B-A D-B-A

Table 3. Data for example 2 Unit

Product

Unit 1 Unit 2 Unit 3 Unit 4

Processing times

A 15 8 12

B 10 20 5 13

C 20 7 9

D 7 17 5

Fig. 5. Schematic diagram for example 2.

Fig. 6 FIS Nw  3 =- vD !# VF 5 4 K @ BR ãä WäM -âò e". ,æ 231 ê

"ë - vM 9Dh ¡=+ ¨  57=# k

 C# "ë ®{.  2D @ G+ vD X9!7 [C vM °M v› JÈ,  D# "ë  2_ vD Y a /G7 [C ¨  57="D vM D' \+, 

 B# v# /Gh± "ë .  4D X9!7 [C Gk v ]h Ž+ °M "ë ®{M 9D @ Ö

=# e ç  !". _ b ¼ C  Y¯ "SK

  FIS Nw -â  !# YZ Ÿ ¨ 

!ë ç  !".

4-3.  3

¼ Ä # Ä 1, 2•" 4YD   6¿,  6¿M  

9»  5  Y¯ K/= •´". (  Table 6R b  VM DhU, @k Úí# Table 7 e /²

". ( v ;< Nw 5 , BR Table 8R 9 -â‚

LU, FIS Nw VF#  5 vD =- !# VF +

=­". Fig. 7 # ZW Nw 5 4K @ BR ãä WäM -â‚".

5.  

IJK. XAW "SK
  Gk v ;<Nw Ox 4K @ A B C ¾K Y¯ w=­+,  Y

¯ / À9»  4Q@yzk 4| }# @

A T=­". †N‰  ‡ˆLM .  / X’“1

 ‘7 8 ( „" "x  c3 A Dr” =# X AW "SK  u1 += À9» v ;< Nw 5

7pLM (( ¨ Å » Ê_ 3K û ü ù  a ¨ /› Ê 0B}# N /²LU,  

¾K Y¯ ÑÒÓÔ{¦ Ô5M ¨=­". w Y¯

ØDh Ä K/ BR ù Y¯ ’/1 •­". ¼ C #  @k  îkR @ pXk µ ªÃ

 îkR pXk µ 7
KLM + Y¯ ¿€ - D” ".

Fig. 6. Result for FIS policy of example 2.

Table 6. Production paths for example 3

Flow characters Production paths

P1 Forward U1→U4→U5→U6

P2 Backward U3→U2→U1

P3 Cycle U3→U5→U2

P4 Forward U3→U4→U5

P5 Cycle U4→U6→U3→U1

P6 Backward U6→U4→U2

Table 7. Data for example 3 Unit

Product

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Processing times

P1 6 101 17 4

P2 8 15 5

P3 20 8 13

P4 9 5 16

P5 151 111 9 7

P6 13 5 101

Table 8. Optimal sequence in each unit for example 3

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6

UIS P1-P5-P2 P2-P6-P3 P2-P3-P5-P4 P5-P6-P1-P4 P3-P1-P4 P6-P5-P1

ZW P1-P2-P5 P2-P3-P6 P2-P3-P5-P4 P5-P1-P6-P4 P3-P1-P4 P5-P6-P1

NIS P1-P2-P5 P2-P6-P3 P2-P3-P5-P4 P5-P6-P1-P4 P3-P1-P4 P6-P5-P1

FIS P1-P2-P5 P2-P6-P3 P2-P3-P5-P4 P5-P6-P1-P4 P3-P1-P4 P6-P5-P1

Table 9. Statistical result for example 3

0-1 Variable Continuous variable Constraint Makespan CPU(sec)

UIS 68 83 263 53 1.15

ZW 68 83 263 59 1.41

NIS 68 83 431 56 1.85

FIS 77 113 501 53 2.17

Fig. 7. Result for ZW policy of example 3.

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