The application of linear programming to a coal-mining problem: A case study

Kle ck, W. G.; Tu ne , G. M.S. A icle The applica ion o linea p og amming o a coal-mining p oblem: A case s udy Sou h A ican Jou nal o Business Managemen P o ided in Coope a ion wi h: Uni e si y o S ellenbosch Business School (USB), Bell ille, Sou h A ica Sugges ed Ci a ion: Kle ck, W. G.; Tu ne , G. M.S. (1985) : The applica ion o linea p og amming o a coal-mining p oblem: A case s udy, Sou h A ican Jou nal o Business Managemen , ISSN 2078-5976, A ican Online Scien i ic In o ma ion Sys ems (AOSIS), Cape Town, Vol. 16, Iss. 1, pp. 46-48, h ps://doi.o g/10.4102/sajbm. 16i1.1072 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/217887 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ The applica ion o linea p og amming o a coal-mining p oblem: A case s udy W.G. Kle ck School o Business Leade ship, Uni e si y o Sou h A ica, P e o ia G.M.S. Tu ne Consul ing enginee This a icle desc ibes he applica ion o linea p og amming o a p oblem aced by a small coal-mining company. The company owns wo ope a ing collie ies, one supplies he gene al inland ade ma ke while he o he supplies a powe s a ion unde a 30-yea - ied con ac . The company was awa ded an expo alloca ion o a 30-yea pe iod and as a esul was ying o es ablish how o supply he coal o he a ious ma ke s. Only he collie y ha supplied he powe s a ion had su icien ese es o base an expo con ac on. I was unde lain by wo economically eco e able seams o coal. The quali y o he coal ese es is, howe e , low compa ed o in e na ional s anda ds and is he e o e unaccep able in i s aw s a e. I is, howe e , possible o upg ade he coal by washing i . The g ade o he inal p oduc ( loa ) can be adjus ed by al e ing he densi y o washing. Howe e he yield, as a pe cen age o he aw coal eed, alls o apidly as he g ade is inc eased. Fu he mo e he sinks can likewise be upg aded by washing. The company wonde ed how hey should op imize he lowshee . How much coal should hey mine om each o he wo seams, how much om each sou ce should go o he powe s a ion, how much should hey p oduce o expo , how should he upg ading be done, and o wha ex en . A linea p og amming model was o mula ed o sol e he p oblem acing he company. S. A . J. Bus. Mgm . 1985, 16: 46- 48 Hie die a ikel besk y die geb uik an linee e p og amme ing om 'n spesi ieke p obleem an 'n klein maa skappy in die s een- koolbed y op e los. Die maa skappy besi wee s eenkoolmyne; een bedien die binnelandse ma k e wyl die ande een 'n k ag- s asie an s eenkool oo sien olgens 'n 30-jaa -kon ak. 'n Ui oe kon ak i 'n pe iode an 30 jaa is aan die maa skappy oeges aan en as ge olg hie an moes die maa skappy ass el op welke wyse die onde skeie ma k e oo sien moes wo d. Slegs die s eenkoolmyn wa die k ags asie bedien he , he genoeg s eenkool ese wes gehad om aan die ui oe kon ak e oldoen. Die myn he wee ekonomieson ginba e s eenkoollae gehad. Die kwali ei an die s eenkool ese wes is eg e laag in e gelyking me in e nasionale s andaa de en die s eenkool moe ge olglik e de e we k wo d. Die e we king geskied deu die on ginde s eenkool e was en sodoende die kwali ei e e hoog. Die kwali ei an die inale p oduk kan e ande wo d deu die dig heid an die wasse y e e s el. Die opb engs, ui ged uk as 'n pe sen asie an die inse , e minde innig as die g aad e hoog wo d. Die a oe (a al) kan ook opgeg adee wo d deu di e was. Die maa skappy wou ass el hoe eel s eenkool an elke s eenkoollaag gemyn moe wo d, hoe eel an elke laag na die k ags asie en hoe eel s eenkool an elke laag na die ui oe ma k moe gaan. Die maa skappy wou ook ass el wa die om ang an die opg ade ing moes wees. 'n Linee e p og amme ingsmodel is geb uik om hie die p obleme an die maa skappy op e los. S.-A . Tydsk . Bed y sl. 1985, 16: 46 -48 W .G. Kle ck* School o Business Leade ship, Uni e si y o Sou h A ica, P.O. Box 392, P e o ia, 0001 Republic o Sou h A ica G.M.S. Tu ne Consul ing enginee , P e o ia *To whom co espondence should be add essed Accep ed Sep embe 1984 In oduc ion This a icle desc ibes a p oblem aced by a small coal-mining company, si ua ed in he eas e n T ans aal, Sou h A ica. The company owns wo ope a ing collie ies. The i s has a p esen capaci y o 1 million ons pe annum and supplies he gene al inland ade ma ke . The second collie y supplies 3,6 million ons pe annum o powe gene a ion unde a 30-yea - ied con ac . The company was ecen ly awa ded an expo alloca ion o a mos 2 million ons pe annum o a 30-yea pe iod. As a esul he company was ying o de e mine wha he mos e icien me hod would be o supply he coal o he a ious ma ke s. Backg ound o he s udy Bo h o he company's collie ies a e unde lain by wo economically eco e able coal-seams, he No. 2 seam and he No. 4 seam. The No. 5 seam, which has me allu gical p ope - ies sui able o he s eel indus y is no p esen while he No. 1 and No. 3 seams a e oo hin o be economically eco e able. The collie y ha supplies he inland ade ma ke has limi ed ese es and has been uled ou by he company as a p oposi ion on which o base an expo collie y. The collie y ha supplies he powe s a ion has subs an ial ese es, in excess o wha is equi ed o he powe s a ion. The powe s a ion supply con ac s ill has 30 yea s o un, he same pe iod as he p oposed expo p ojec . In o ma ion on ese es and a e o exploi a ion we e ob ained and a e shown in Table 1. In addi ion o hese ese es he company has access o a mini-pi p oposi ion. This could, i necessa y, supply an addi ional 0,4 million ons pe annum o he No. 4 seam o he du a ion o he p ojec . Table 1 Rese es and a e o exploi a ion Remaining eco e able ese e (million ons) Requi ed li e (yea s) Maximum a e o exploi- a ion/yea (million ons) Quali y o Sou h A ican coal No. 2 seam No. 4 seam 90 30 3,0 99 30 3,3 Sou h A ican coal ese es a e gene ally o low quali y by in e na ional s anda ds and as such a e unaccep able o s. A . J. Bus. Mgm . 1985, 16(1) expo in hei aw s a e. In his pa icula case he quali y o he No. 2 seam is 25,5 MJ/Kg and ha o he No. 4 seam 22,0 MJ/Kg, while he minimum in e na ional-accep able quali y is 27 MJ/Kg. (MJ/Kg o megajoules pe kilog am is a measu e o he hea con en o calo i ic alue o he coal. The highe he hea con en he mo e accep able he coal.) Howe e , i is possible o upg ade he coal by washing. The aw coal is c ushed and hen ed in o a essel con aining a hea y liquid medium. This sepa a es he coal in o wo s eams, wi h pa o he coal loa ing and pa sinking. The coal pa icles ha loa ha e a highe calo i ic alue han he aw-coal eed, while he pa icles ha sink a e o a lowe g ade. The g ade o he loa s can be adjus ed by al e ing he densi y o washing. Howe e he yield, as a pe cen age o he aw-coal eed, apidly alls o as he calo i ic alue is aised, and he p ocess becomes uneconomical. An analysis was done o he washing cha ac e is ics o he company's coal. The esul s showed a e y poo yield o he No. 4 seam and con i med ha i was no a p oposi ion o upg ade his yield o expo -quali y coal. The No. 2 seam in con as showed good washing cha ac e is ics. The No. 2 seam was es ed a wo di e en densi ies, o p oduce a 27 MJ/Kg and a 28 MJ/Kg loa p oduc (Table 2). The company doub ed he economics o he 28 MJ/Kg p oduc because o low yield. Table 2 Yield o No. 2 seam Densi y I Densi y 2 P oduc quali y, MJ/Kg 27 28 P oduc yield ( loa s), OJo 79 68 Sinks, OJo 21 32 Sinks quali y, MJ/Kg 19,5 22,3 In he same way ha aw coal can be washed, he, sinks can be upg aded by washing a a highe densi y in a second- s age p ocess, o o m a middlings p oduc . The analysis o he washing cha ac e is ics o he second s age is showed in Table 3. The quali y o he disca ds om he second s age would no be accep able o use and would ha e o be dumped. The washing plan has a i s -s age eed capaci y o 2,8 million ons pe annum wi h a second-s age capaci y o 1, 1 million ons pe annum inpu . Table 3 Analysis o second-s age washing Feed quali y, MJ/Kg Middlings yield, OJo Middlings quali y, MJ/Kg Disca d, OJo Disca d quali y, MJ/Kg The powe -s a ion con ac Densi y I 19,5 56 23,9 44 12 Densi y 2 22,3 75 25,I 25 12 When he c.:ompany o iginally ende ed o he powe -s a ion supply i IJlanned o mine he No. 2 seam and he No. _4 seam o his ,u pose and blend hem oge he . (Coals o di e en calo i ,V.: alues can be blended linea ly.) The company had ende ed o a minimum calo i ic alue o 23 MJ /Kg o he 47 li e o he s a ion. The powe s a ion had acco dingly been designed o his minimum and he company was con ac ually commi ed o supply his quali y. Any de e io a ion would se e ely a ec powe -s a ion e iciency and ou pu and could no be con empla ed. P oduc ion cos s and sales e enue The No. 4 seam is no as ha d bu hicke han he No. 2 seam and cos s R5,30/ on o mine compa ed o R7,50/ on o he la e . The mini-pi mining cos s had been calcula ed a R9,30/ on, including o e land anspo . The cos o ope a ing he washing plan was es ima ed a Rl,40/ on o eed o he plan wi h second-s age cos s a R0,80/ on inpu . On he e enue side he con ac ual p ice o eed o he powe s a ion was R9,90/ on. Expo coal could be sold o R40/ on FOB o he 27 MJ/Kg p oduc . A e allowing o ailage, po , and handling cha ges o Rl4,40/ on, his le he company wi h an income o R25,60/ on. Simila ly, o a 28 MJ/Kg p oduc he income was calcula ed a R27, 70/ on. Issues acing he company Al hough he No. 4 seam and mini-pi we e only sui able o powe -s a ion eed, he No. 2 seam which had adi ionally been sen o he powe s a ion o make up he equi ed calo i ic alue, could now be washed a ei he o wo densi ies o p oduce a 27 MJ/Kg and/o 28 MJ/Kg o expo , as well as being bypassed o he powe s a ion as be o e. The sinks om ei he o hese ope a ions could be sen o he powe s a ion o al e na i ely could be washed in he second s age o upg ade he quali y, wi h he middlings ou ed o he powe s a ion. The company wonde ed how hey should go abou op imizing he low-shee . How much coal should hey mine om each o he No. 2 seam, No. 4 seam, and mini-pi ? How much om each sou ce would go di ec ly o he powe s a ion? How much should hey p oduce o expo and wha quali y should be p oduced? To wha ex en should hey use he second-s age washing p ocess, conside ing i s low yield? Rese es did no appea o be a cons ain wi h a mining capaci y o 6, 7 million ons pe annum. Nei he did he capaci y o he mine's sha o 5,8 million ons pe annum appea o be a cons ain . Bu would he washing plan be big enough? The model The p oblem was o mula ed as a linea p og amming model. To o mula e he model a low cha o he 'mining ope a ions' was d awn. This is shown in Figu e 1. Decision a iables The decision a iables appea on he low cha and a e: S1 Tons mined pe annum om he mini-pi . S2 -Tons mined pe annum om No. 4 seam. S3 -Tons mined pe annum om No. 2 seam. W1 -No. 2 seam ons pe annum o powe s a ion. W2 -No. 2 seam ons pe annum o Washing S age 1, Densi y 1. W3 -No. 2 seam ons pe annum o Washing S age l, Densi y 2. X1 -Tons o 27 MJ/Kg quali y expo ed pe yea . X2 -Tons o 28 MJ/Kg quali y expo ed pe yea . Y1 -Tons sinks pe annum om Washing S age l, Densi y 1 o Powe S a ion. 48 s. (22.0 MJ/Kg) s, (22.0 MJ/Kg) w. No. 4 X, seam Wash s, w, S age 1 Collie y No. 2 Densi y 1 Seam (25,5 MJ/Kg) x, Y, Sinks 0,21 ' Y, Y, Sinks Middlings 0.56 Y, Wash S age 2 Densi y 1 S.-A . Tydsk . Bed y sl. 1985, 16(1) P ( o al eed) Powe S a ion 0.44 Y, Middlings Expo Minipi Wash 0,32 w, Y, Wash 0,75 Y, S age 1 S age 2 Densi y 2 Densi y 2 0,25 Y, Figu e 1 F1ow diag am o mining ope a ions. Y2 -Tons sinks pe annum om Washing S age l, Densi y 1 o Washing S age 2, Densi y I. Y3 -Tons sinks pe annum om Washing S age l, Densi y 2 o Powe S a ion. Y4 -Tons sinks pe annum om Washing S age 1, Densi y 2 o Washing S age 2, Densi y 2. P -To al eed o he powe s a ion. The cons ain s o his p oblem e e o he maximum a e o exploi a ion, sha capaci y, ail con ac , expo con ac , washing-plan capaci y, powe -s a ion supply, and he g ade cons ain o he powe -s a ion supply. These cons ain s as well as he objec i e unc ion a e as ollows: Ma e ial balance S3 - Wi - W2 - W3 = 0 Xi - 0, 79 W2 = 0 0,21 W2 - Yi - Y2 = 0 X2 -0,68 W3 = 0 0,32 W3 - Y3 - Y4 = 0 Si + S2 + W, + Yi + 0,56 Y2 + Y3 + 0, 75 Y4 -P = 0 Maximum a e o exploi a ion s. ~ 0,4 S2 ~ 3,3 S3 ~ 3,0 Sha capaci y Si + S3 ~ 5,8 Rail con ac Xi + X2 e; 1,0 Expo con ac Xi + X2 ~ 2,0 Washing-plan capaci y W2 + W3 ~ 2,8 Y2 + Y4 ~ 1,1 Powe -s a ion supply P = 3,6 G ade cons ain powe s a ion To al MJ p oduced ~ 23 0 To al ons p oduced - ' i.e. 22,0 S1 + 22,0 S2 + 25,5 W, + 19,5 Yi + 22,3 Y3 + 23,9 (0,56 Y2) + 25,1 (0,75 Y4) e; 23(3,6) Objec i e unc ion Maximize -9,3 S1 -5,3 S2 -7,5 S3 -1,4 W2 -1,4 W3 -0,8 Y2 -0,8 Y4 + 9,9 P + 25,6 Xi + 27,7 X2 Resul s The solu ion ob ained om he model indica es ha he company should mine 3 million ons pe annum om he No. 2 seam and 2,5 million ons om he No. 4 seam pe annum. No hing should be mined om he mini-pi . O he 3 million ons mined om he No. 2 seam app oxima ely 0,5 million ons should be sen di ec ly o he powe s a ion and 2,5 million ons o he washing plan whe e Densi y-2 washing will be applied. App oxima ely 1, 7 million ons o his 2,5 million ons will be expo ed, he quali y being 28 MJ/Kg. Du ing S age 2 o he washing plan 0,8 ons will be p ocessed. O he 0,8 ons app oxima ely 0,6 ons will be ou ed o he powe s a ion. The a e age quali y o he coal supplied o he powe s a ion is 23 MJ/Kg. The o al yea ly income ecei ed by ollowing his solu ion is R42,9 million. Wi h he excep ion o he powe s a ion g ade cons ain and he No. 2 seam a e o exploi a ion cons ain all he cons ain s ha e slack; indica ing ha he capaci ies o he sha , wo king plan , e c. a e su icien . Acco ding o he solu ion he company should supply l, 7 million o he 2 million ons expo con ac . The eason being ha he No. 2 seam maximum a e o exploi a ion cons ain is binding and he expo con ac is only supplied om his seam. I he company wan s o inc ease he onnage on he expo con ac i will ha e o conside i s 'policy on he maximum a e o exploi a ion. Conclusion ' ' This case s udy desc ibed he applica ion ·~ linea p og amming o a small coal-mining p oblem and i )us a ed how linea p og amming can assis he manageme H.. o he company o sol e a easonable complex p oblem.