Kostenoptimierung für Baumwollmischungen: Ansatz der Fuzzy lineare Programmierung

24 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. 25 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. 26 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. 27 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 28 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 29 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.