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ACC JOURNAL 2015, Volume 21, Issue 1 DOI: 10.15240/ ul/004/2015-1-003
OPTIMIZATION OF COTTON MIXING COST: A FUZZY LINEAR PROGRAMMING
APPROACH
Subhasis Das1; Anindya Ghosh
Go e nmen College o Enginee ing & Tex ile Technology,
Depa men o Tex ile Technology, Be hampo e, India
e-mail: 1[email p o ec ed]
Abs ac
This pape deals wi h he minimiza ion o he cos o h ee ypes o co on mixings sui able
o he p oduc ion o ine, medium and coa se ya ns wi hou impinging he equi ed quali y
using he uzzy linea p og amming, which is he amalgama ion o uzzy logic and classical
linea p og amming. The indings o his wo k poin ou ha he uzzy linea p og amming
a ains he bes solu ion o co on mix in compa ison wi h i s classical e sion as well as he
exis ing me hod ollowed in co on spinning indus y.
In oduc ion
The concep o uzzy linea p og amming was i s p oposed by Bellman and Zadeh [1].
The ea e , Zimme mann [2], Ve degay [3, 4], Chanas [5], We ne s [6], Tan [7] e c. made
signi ican con ibu ions in he de elopmen o uzzy linea p og amming. Elam azu hi e al.
[8] used uzzy linea p og amming in p oduc ion planning in ex ile indus y.
Judicious mixing o di e en ypes o co on ib es is an op imiza ion p oblem o ob aining
aw ma e ial wi h equi ed quali y a a minimum cos in he co on spinning indus ies. In his
wo k an a emp has been made o sol e he co on mixing p oblem using uzzy linea
p og amming app oach. The case s udies ha e been made on h ee ypes o co on mixings
sui able o he p oduc ion o ine, medium and coa se ya ns. The da a o his p oblem a e
aken om he co on spinning indus y. In keeping wi h he adi ional indus ial p ac ice,
wo a ie ies o co ons a e conside ed o each ype o mixing. A compa a i e s udy o he
ela i e pe o mances o uzzy linea p og amming and i s classical e sion is p esen ed he e.
I also a emp s o es ablish i s e ec i eness o e he usual p ac ice as ollowed by he co on
spinning indus y.
1 Op imiza ion o Co on Mixing Cos
Co on ib e mixings sui able o he p oduc ion o ine, medium and coa se ya ns a e
conside ed in his s udy, and he da a we e collec ed om he co on spinning indus y. Fo
each ype o mixing wo a ie ies o co ons a e used. The objec i e o his s udy is o
de e mine he p opo ion o co ons o each ype o mixing o achie e he bes quali y ya ns
a he lowes aw ma e ial cos . Judging om heo y and a p ac ical s andpoin , ou ib e
p ope ies, such as ib e bundle s eng h, uppe hal mean leng h (UHML), ib e ineness and
sho ib e con en (SFC) a e conside ed since hey exe a s ong in luence on he ya n
quali y. Table 1 depic s he ib e p ope ies and cos s o di e en ypes o co ons.
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Tab. 1: Fib e p ope ies and cos o di e en ype o co on mixing
Mixing
ype
Co ons
used
Bundle
s eng h
(cN / ex)
UHML
(mm)
Fineness
(µg/inch)
Sho ib e
con en
(%)
Cos / 100 Kg
co ons
(USD)
Fine
ULTIMA
34.5
31.80
3.9
20.90
394.40
PIMA
38.8
34.30
4.2
12.50
367.00
Medium
DCH -32
39.1
35.12
4.2
16.10
291.40
MCU-5
31.5
31.10
4.3
21.50
269.53
Coa se
J-34
32.1
28.73
4.8
19.70
268.80
S-6
31.3
28.17
4.4
20.16
281.64
Sou ce: Own
1.1 Fuzzy Linea P og amming App oach
As he bundle s eng h and UHML a e bene i c i e ia whe e highe alues a e desi able, and
ineness and SFC a e cos c i e ia whe e lowe alues a e be e , he p oblem o each ype o
co on mixing can be o mula ed as he ollowing uzzy linea p og amming p oblem:
Minimize 𝑐1𝑥1+ 𝑐2𝑥2 (mixing cos )
Subjec o,
𝑎11𝑥1+ 𝑎12𝑥2≥ 𝐸1 (s eng h)
𝑎21𝑥1+ 𝑎22𝑥2≥ 𝐸2 (UHML)
𝑥1
𝑎31
+𝑥2
𝑎32
≥1
𝐸3 ( ineness)
𝑎41𝑥1+ 𝑎42𝑥2≤ 𝐸4 (SFC)
𝑥1+ 𝑥2= 1
𝑥1, 𝑥2≥ 0
whe e x1 and x2 a e he p opo ions o wo a ie ies o co ons, c1 and c2 a e hei
co esponding cos s, E1, E2, E3 and E4 ep esen he mono onically dec easing membe ship
unc ion o he cons ains such as ib e bundle s eng h, UHML, ineness and SFC,
espec i ely.
The objec i e unc ions and cons ain equa ions o ine, medium and coa se co on mixings
a e o mula ed in Table 2. The membe ship unc ions o he a ious cons ain s o a
pa icula mixing a e de e mined by ansla ing ou pe cep ion and expe ience om he
a ailable ib e da a. The ob ious equali y cons ain in he p oduc mix p oblem is he sum o
x1 and x2 equals o one. MATLAB 7.11 coding has been used o sol e he p oblems.
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Tab. 2: Objec i e unc ions, uzzy numbe s and cons ain s o uzzy linea p og amming o
di e en co on
Mixing ype
Fine
Medium
Coa se
Objec i e
Func ion
394.4𝑥1 + 367.0𝑥2
291.4𝑥1 + 269.53𝑥2
268.8𝑥1+ 281.64𝑥2
E1
1 when 𝑥 ≤ 28.5
38.5 − 𝑥
38.5 − 29.5 when 28.5 < 𝑥 ≤ 38.5
0 when 38.5 < 𝑥
1 when 𝑥 ≤ 27.5
38.0 − 𝑥
38.0 − 27.5 when 27.5 < 𝑥 ≤ 38
0 when 38.0 < 𝑥
. 0
1 when 𝑥 ≤ 28.5
32.0 − 𝑥
32.0 − 28.5 when 28.5 < 𝑥 ≤ 32.0
0 when 32.0 < 𝑥
E2
1 when 𝑥 ≤ 27.5
35.2 − 𝑥
35.2 − 27.5 when 27.5 < 𝑥 ≤ 35.2
0 when 35.2 < 𝑥
1 when 𝑥 ≤ 27.5
35.0 − 𝑥
35.0 − 27.5 when 27.5 < 𝑥 ≤ 35.0
0 when 35.0 < 𝑥
1 when 𝑥 ≤ 26.5
29.0 − 𝑥
29.0 − 26.5 when 26.5 < 𝑥 ≤ 29.0
0 when 29.0 < 𝑥
E3
1 when 1
𝑥≤1
4.3
1
3.5 −1
𝑥
1
3.5 −1
4.3
when 1
4.3 <1
𝑥≤1
3.5
0 when 1
3.5 <1
𝑥
1 when 1
𝑥≤1
4.8
1
3.7 −1
𝑥
1
3.7 −1
4.8
when 1
4.8 <1
𝑥≤1
3.7
0 when 1
3.7 <1
𝑥
1 when 1
𝑥≤1
4.9
1
3.9 −1
𝑥
1
3.9 −1
4.9
when 1
4.9 <1
𝑥≤1
3.9
0 when 1
3.9 <1
𝑥
E4
1 when 𝑥 ≤ 10.0
16.0 − 𝑥
16.0 − 10.0 when 10.0 < 𝑥 ≤ 16.0
0 when 16.0 < 𝑥
1 when 𝑥 ≤ 15.5
20.5 − 𝑥
20.5 − 15.5 when 15.5 < 𝑥 ≤ 20.5
0 when 20.5 < 𝑥
1 when 𝑥 ≤ 18.0
20.5 − 𝑥
20.5 − 18.0 when 18.0 < 𝑥 ≤ 20.5
0 when 20.5 < 𝑥
Equali y
cons ain s
𝑥1+ 𝑥2= 1
𝑥1+ 𝑥2= 1
𝑥1+ 𝑥2= 1
Va iable
bounda y
𝑥1, 𝑥2≥ 0
𝑥1, 𝑥2≥ 0
𝑥1, 𝑥2≥ 0
Sou ce: Own
1.2 Linea P og amming App oach
The same objec i e unc ions a e also sol ed using he linea p og amming p oblem which is
o mula ed as:
minimize 𝑐1𝑥1+ 𝑐2𝑥2
subjec o,
𝑎11𝑥1+ 𝑎12𝑥2≥ 𝑒1 (s eng h)
𝑎21𝑥1+ 𝑎22𝑥2≥ 𝑒2 (UHML)
𝑥1
𝑎31
+𝑥2
𝑎32
≥1
𝑒3 ( ineness)
𝑎41𝑥1+ 𝑎42𝑥2≤ 𝑒4 (SFC)
𝑥1+ 𝑥2= 1
𝑥1, 𝑥2≥ 0
Cons ain s o linea p og amming p oblem o di e en co on mixings a e abula ed in
Table 3.
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Tab. 3: Cons ain s o linea p og amming p oblem o di e en co on mixings
Mixing ype
Fine
Medium
Coa se
Inequali y
cons ain s:
S eng h
34.5𝑥1+38.8𝑥2≥36.5
39.1𝑥1+31.5𝑥2≥35.0
32.1𝑥1+31.3𝑥2≥31.6
UHML
31.8𝑥1+34.3𝑥2≥33.1
35.12𝑥1+31.1𝑥2≥33.1
28.73𝑥1+28.17𝑥2≥28.1
Fineness
𝑥1
3.9 +𝑥2
4.2 ≥1
4.1
𝑥1
4.2 +𝑥2
4.3 ≥1
4.25
𝑥1
4.8 +𝑥2
4.4 ≥1
4.6
SFC
20.9𝑥1+12.5𝑥2≤17.0
16.1𝑥1+21.5𝑥2≤18.5
19.7𝑥1+20.16𝑥2≤20.5
Equali y cons ain s
𝑥1+ 𝑥2= 1
𝑥1+ 𝑥2= 1
𝑥1+ 𝑥2= 1
Va iable bounda y
𝑥1, 𝑥2≥ 0
𝑥1, 𝑥2≥ 0
𝑥1, 𝑥2≥ 0
Sou ce: Own
The pe o mance o uzzy linea p og amming o op imizing co on mixing cos has been
compa ed wi h ha o linea p og amming is-à- is he exis ing mill p ac ice. Table 4
illus a es he op imum p opo ion o co on mixing as well as he mixing cos as ob ained by
di e en app oaches.
Tab. 4: Mixing cos as ob ained by di e en me hod
Mixing
ype
Co ons used
Indus y p ac ice
Linea p og amming
Fuzzy linea
p og amming
Mixing
p opo ion
Mixing
cos
($/100kg)
Mixing
p opo ion
Mixing
cos
($/100kg)
Mixing
p opo ion
Mixing
cos
($/100kg)
Fine
ULTIMA/PIMA
50/50
380.70
32/68
375.76
27/73
374.73
Medium
DCH-32/ MCU-5
60/40
282.65
56/44
281.68
37/63
277.53
Coa se
J-34/S-6
50/50
275.22
52/48
274.93
81/19
271.29
Sou ce: Own
The ob ained p ope ies o h ee ypes o co on mixings sol ed by bo h uzzy linea
p og amming and linea p og amming a e gi en in Table 5. I is obse ed om Tables 4 and 5
ha he uzzy linea p og amming in a iably p o ides he bes solu ion o he mixing
p opo ion by ul illing he condi ion o all cons ain s o ensu e he leas mixing cos . As an
example, i is e iden om Table 4 ha in he case o ine mixing, he mixing cos pe 100 kg
is educed om $380.70 o $375.76 while he p oblem is sol ed by linea p og amming and
he mixing cos is u he cu bed down o $374.73 while i is sol ed by uzzy linea
p og amming. I hus ollows ha a mix o 32/68 o ULTIMA and PIMA co ons as ob ained
by linea p og amming p oduces a cheape mixing han he 50/50 mix as p ac iced by he mill
o a ine mixing, u he a mix o 27/73 as ob ained by he uzzy linea p og amming yields a
minimum mixing cos .
Tab. 5: Ob ained p ope ies o he co on mixing
Mixing
ype
Bundle
s eng h
(cN / ex)
Uppe hal
mean leng h
(mm)
Fineness
(mic onai e
alue)
SFC
(%)
Fuzzy linea
p og amming
Fine
37.66
33.67
4.12
14.79
Medium
34.28
32.57
4.26
19.53
Coa se
31.94
28.62
4.72
19.79
Linea
p og amming
Fine
37.44
33.53
4.10
15.16
Medium
35.72
33.33
4.24
18.50
Coa se
31.72
28.46
4.60
19.92
Sou ce: Own
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A simila kind o obse a ion is no ed in he case o medium and coa se co on mixings as
shown in Table 4. The dominance o uzzy linea p og amming o he co on mixing p oblem
may be a ibu able o i s abili y o handling he si ua ion whe e he e a e no exac de ined
bounda ies o he inequali y cons ain s. Fo example, a spinne o en uses he e ms such as
low and high o assess he ib e s eng h, leng h, ineness and SFC e c., howe e hese e ms
do no cons i u e a well-de ined bounda y. All hese ib e p ope ies ha e app oxima e
bounda ies a he han exac bounda ies. In addi ion, he membe ship unc ions o he a ious
cons ain s could be de eloped by deciphe ing he expe ience o a spinne which ende s a
be e u iliza ion o he ib e p ope ies needed o a pa icula mixing. The e o e, uzzy linea
p og amming sa is ies bo h he goal and cons ain s wi h a maximum deg ee.
Conclusion
Fuzzy linea p og amming is used o op imize he aw ma e ial cos pe aining o h ee co on
mixings sui able o he p oduc ion o ine, medium and coa se ya ns. As pe equen ly
ollowed indus ial p ac ice, wo a ie ies o co ons a e conside ed o each ype o mixing.
The solu ion ob ained by uzzy linea p og amming o he co on mixing p oblem has been
compa ed wi h ha o linea p og amming as well as he exis ing sys em as p ac ised by he
co on spinning indus y. Fuzzy linea p og amming eme ges as he mos po en app oach in
his ega d o all ypes o mixings ollowed by he linea p og amming. Fuzzy linea
p og amming can enable o handle he imp ecision ha is p esen in he inequali y cons ain s
o ib e s eng h, leng h, ineness and SFC. A diminu i e educ ion in he aw ma e ial cos
pe 100 kg by e aining he ib e quali y a a equisi e le el may lead o a conside able amoun
o economic gain o he co on spinning indus y. The e o e, he uzzy linea p og amming
app oach o selec ion o he co on mix has an eno mous scope o indus ial accep ance.
Li e a u e
[1] BELLMAN, R. E.; ZADEH, L.A.: Decision Making in a Fuzzy En i onmen .
Managemen Science. 1970, 17, 141-164.
[2] ZIMMERMANN, H.J.: Fuzzy Se s & Sys ems. 1978, 1, 45-55.
[3] VERDEGAY, J. L.: Fuzzy ma hema ical p og amming. In: Gup a, M. M. and Sanchez,
E. (eds.) Fuzzy In o ma ion and Decision P ocesses. Ams e dam, 1982.
[4] VERDEGAY, J. L.: Fuzzy Se s & Sys ems. 1984, 1, 131-141.
[5] CHANAS, S.: Fuzzy se s & Sys ems. 1983, 11, 243-251.
[6] WERNERS, B.: Eu opean Jou nal o Ope a ions Resea ch. 1987, 31, 342-349.
[7] TAN, R. R.: En i onmen al Modeling & So wa e. 2005, 20, 1343-1346.
[8] ELAMVAZUTHI, I.; GANESAN, T.; VASANT, P.; WEBB, F. J.: In e na ional
Jou nal o Compu e Science and In o ma ion Secu i y. 2009, 6, 238-243.
Subhasis Das; Anindya Ghosh

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OPTIMALIZACE CENY BAVLNĚNÝCH SMĚSÍ: PŘÍSTUP FUZZY LINEÁRNÍHO
PROGRAMOVÁNÍ
Ten o článek se zabý á minimalizací nákladů u ří ypů ba lněných směsí hodných p o
ý obu jemné, s řední a h ubé příze, aniž by byla oh ožena požado aná k ali a, pomocí z .
uzzy lineá ního p og amo ání, k e é je sloučením uzzy logiky a klasického lineá ního
p og amo ání. Zá ě é o p áce poukazuje na o, že uzzy lineá ní p og amo ání dosahuje
u ba lněných směsí lepšího ýsledku e s o nání s klasickou e zí, jakož i se s á ajícími
me odami použí anými při p ůmyslo ém předení ba lny.
KOSTENOPTIMIERUNG FÜR BAUMWOLLMISCHUNGEN: ANSATZ DER FUZZY
LINEARE PROGRAMMIERUNG
Diese A ikel be ass sich mi de Kos enminimie ung bei d ei Typen on
Baumwollmischungen bei gleichbleibende Quali ä , die zu He s ellung eine , mi le e und
g obe Ga ne geeigne sind. Dies geschieh mi Hil e des so genann en Fuzzy Linea e
P og ammie ung, welches in einem Zusammenschluss on Fuzzy-Logik und dem klassischen
linea en P og ammie en bes eh . Die E gebnisse diese A bei zeigen, dass das Fuzzy Linea e
P og ammie ung bei Baumwollmischungen im Ve gleich mi de klassischen Ve sion und mi
den bes ehenden, beim indus iellen Ve spinnen de Baumwolle angewand en Me hoden
besse e E gebnisse au weis .
OPTYMALIZACJA CENY MIESZANEK BAWEŁNY: ZASTOSOWANIE ROZMYTEGO
PROGRAMOWANIA LINIOWEGO
Niniejszy a ykuł poświęcony jes zagadnieniu minimalizacji kosz ów w p zypadku zech
odzajów mieszanek bawełny nadających się do p odukcji cienkiej, ś edniej i g ubej p zędzy
p zy zachowaniu wymaganej jakości, z zas osowaniem me ody zw. ozmy ego
p og amowania liniowego ( uzzy linea p og amming), k ó a jes połączeniem logiki ozmy ej
i klasycznego p og amowania liniowego. W zakończniu niniejszego op acowania wskazano,
że ozmy e p og amowanie liniowe daje w p zypadku mieszanek bawełny lepsze e ek y
w po ównaniu z we sją klasyczną, jak i is niejącymi me odami s osowanymi
w p zemysłowym p zędzeniu bawełny.